diff --git "a/99FPT4oBgHgl3EQfZDSN/content/tmp_files/load_file.txt" "b/99FPT4oBgHgl3EQfZDSN/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/99FPT4oBgHgl3EQfZDSN/content/tmp_files/load_file.txt" @@ -0,0 +1,859 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf,len=858 +page_content='EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We generate anti-self-polar polytopes via a numerical implementation of the gradient flow induced by the diameter functional on the space of all finite subsets of the sphere, and prove related results on the critical points of the diameter functional as well as results about the combinatorics of such polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We also discuss potential connections to Borsuk’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Pointwise extremal sets 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The pyramid construction 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Construction of k-stacks 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Minimal sets on S2 with diameter below the first accumulation critical value 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Configuration space 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Finiteness results 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A labeling strategy for the points in Bk 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Anti-self-polar polytopes 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' ASP polytopes 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Borsuk’s conjecture 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof of Lovasz’s theorem 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 4-dimensional polytopes 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Implementation of the diameter gradient flow 16 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Computational results 17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Pointwise extremal configurations on S2 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Pointwise extremal configurations on S3 20 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Semi-algebraic sets 21 References 21 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Introduction Let (X, dX) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The Kuratowski embedding x �→ dX(x, ·) is an embedding of X into L∞(X), the space of all bounded real-valued functions on X with the uniform norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When X is the unit sphere with its geodesic distance, the homotopy types of the r-neighborhoods Br(X, L∞(X)) in the Kuratowski embedding of X were studied by Katz in [Kat91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The values at which the homotopy type changes are closely related to the critical configurations of the diameter functional diam of X which maps a finite subset A of X to diam(A) := maxa,a′∈A dX(a, a′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When X is the unit circle, such critical values turn out to be exactly one-half of the diameter values of odd regular polygons inscribed in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Note that 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='13076v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='CO] 30 Jan 2023 2 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG the vertex sets of odd regular polygons are exactly the configurations that are local minima of the diameter functional on the space of all finite subsets of S1 equipped with Hausdorff distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In [Kat89], Katz studied the diameter-extremal configurations on S2 and S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The latter provide candidates for testing Borsuk’s conjecture in R4 (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Recently, Lim, M´emoli, and Okutan [LMO22, Theorem 5] proved that the homotopy types of neighborhoods of the Kuratowski embedding of X are naturally homotopy equivalent to the so-called Vietoris–Rips complexes of X, a central object in the field of applied algebraic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Therefore, the study of diameter-extremal configurations is also of interest for understanding the properties of the Vietoris-Rips complex of spheres [AA17, AAF18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In this paper, we extend the investigation of diameter-extremal configurations on spheres started in [Kat89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In the S1 case, the critical values of the diameter functional form a convergent sequence with the only accumulation point being π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' It is natural to wonder to what extent a similar behavior is true on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We consider two canonical families of diameter- extremal configurations on S2 which we call pyramids Ak that contains 2k + 2 points (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1) and stacked-triangles Bk that contains 3k + 1 points (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Both families contain infinitely many members with diameters monotonically approaching 2π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We prove in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10 that 2π 3 is in fact the first accumulation point of the set of critical values of the diameter functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='17, we prove that the two families Ak and Bk do not exhaust all the possible configurations with similar diameter bounds, and in fact there are infinitely many additional diameter-extremal configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Diameter-extremal configuration with 3k points can be found by performing diameter gradient flow on a certain subset of Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When k is odd, by a parity argument, the resulting configuration cannot be an instance of Ak or Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We next devise and implement a computational algorithm (see Algorithm 1) that attempts to produce diameter extremal configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We use this algorithm to find new configu- rations not in Ak or Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Furthermore, we found configurations not isometric to the ones produced in the course of proving Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' See Table 1 for a complete list of all the configurations we found in this way with up to 10 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The list contains 10 previously unknown configurations where 8 of those exhibit Z2 symmetry and the remaining two are asymmetric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' see Figures 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The convex hulls of certain diameter-extremal configurations give rise to anti-self-polar polytopes (ASP), for example, the regular tetrahedron and any Ak or Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' ASPs are polytopes P characterized by the property that the polar of P equals −P (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' ASPs have been studied by Lov´asz in the context of answering a question by Erd¨os and Graham [Lov83] and were also considered in [Kat89, Section 5] in the context of Borsuk’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Borsuk’s conjecture (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2) for a finite point set X in Rn is equivalent to the property that the chromatic number of the diameter graph (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7) of X is bounded above by n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We continue to explore the suggestion in [Kat89] to use diameter- extremal configurations on S3 to test Borsuk’s conjecture in R4 (a case that is still open).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As shown by Lovasz [Lov83], the chromatic number of the diameter graph associated to any ASP in Rn is at least n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' An ASP for which the inequality is strict would disprove Borsuk’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' It was conjectured in [Kat89] that the number of edges in the diameter graph of an ASP 4-polytope with v vertices is at least 3v − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We use Kalai’s inequality from [Kalai94, Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3] to prove such a bound in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='21 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We then formulate conjectures EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 3 about the number of edges in the diameter graph for more general subsets on S3, see Conjec- tures 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='22 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A calculation based on these two conjectures suggests that the maximum possible chromatic number of the diameter graph of a finite subset X ⊆ R4 is 6 instead 5, the number predicted by Borsuk’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We perform experiments attempting to identify diameter-extremal configurations on the three-dimensional sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The interest in these experiments is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' On the one hand, it is naturally interesting to obtain an understanding of critical configurations beyond the case of S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' On the other hand, whereas Borsuk’s conjecture is known to be true in dimensions 2 and 3 but false in dimensions 64 and higher, its status for dimension 4 is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Hence, by the above, it is tempting to seek a diameter-extremal configuration X of S3 whose convex hull is an ASP such that its diameter graph has chromatic number at least 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We discovered 65 new configurations on S3 not obtained by the pyramid construction (see 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1) on a previously known configuration on S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' However, all the diameter graphs of these configurations have a chromatic number precisely 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' This work was partially supported by BSF #2020124, NSF CCF #1740761, and NSF IIS #1901360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Pointwise extremal sets Let Sn ⊆ Rn+1 be the unit sphere with its geodesic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For a subset Y ⊆ Sn, its diameter diam(Y ) is computed with respect to the geodesic distance on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1 (Taut sets in Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A finite subset Y ⊂ Sn is taut if one of the following equivalent conditions is satisfied: (1) the convex hull of Y contains the origin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' (2) there are non-negative real numbers {ay}y∈Y , not all zero, satisfying � y∈Y ay y = 0, where y denotes the position vector of the point y ∈ Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Jung’s theorem immediately gives the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If Y ⊂ Sn is taut, then diam(Y ) ≥ arccos � −1 n+1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The following observation will be useful in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ⊂ Sn be a taut set such that |Y | = n+2 and diam(Y ) < arccos � − 1 n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then the dimension of the vector space spanned by Y is equal to n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In particular, if {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , an+2} is any set of non-negative coefficients such that n+2 � i=1 aiyi = 0, then all ai must be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Suppose the vector space spanned by all points in Y is of dimension at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then, the set {y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , yn+2} must lie on some great sphere Sn−1 ⊆ Sn and it must be taut in Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2, the set Y must have diameter at least arccos � − 1 n � which contradicts the assumptions on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' This concludes the first part of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For the second part, without loss of generality, we assume that a1 = 0, then the set of vectors 4 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG {y2, y3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , yn+2} is linearly dependent and hence dim(span{y2, y3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , yn+2}) < n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The contradiction with the first part establishes the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ Let Y be a subset of a metric space (X, dX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For any two points y, y′ ∈ Y , we say that y and y′ are comaximal in Y if dX(y, y′) = diam(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In such a case, y is called a comaximal point with y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We use the notation comaxY (y) to denote the set of all points in Y which are comaximal with y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For two points x, x′ ∈ Sn with distance less than π, there is a unique arclength-parametrized geodesic γx,x′ connecting x to x′ such that γx,x′(0) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Consider the unit tangent vector ˙γx,x′(0) in the tangent space TxSn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We recall the notion of pointwise extremal subsets in Sn as in [Kat89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4 ([Kat89]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ⊆ Sn be a finite subset with no antipodal pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We say that y ∈ Y is held (in place) by Y if the set of vectors ˙γy,y′(0) as y′ runs over comaxY (y) is a taut set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We say that Y is pointwise extremal if every point y ∈ Y is held by Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When n = 1, it is not difficult to see that, for all integers k ≥ 1, the vertex set of an inscribed regular (2k + 1)-gon is pointwise extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The following proposition shows the converse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ⊆ S1 be a pointwise extremal set containing no pair of antipodal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then Y is the vertex set of an odd regular polygon inscribed in S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let y ∈ Y and let D = diam(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let RD be the clockwise rotation on S1 by angle D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As y is held by Y ⊆ S1, the set Y must contain both points in S1 at distance D from y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In particular, the set Y is invariant under the rotation RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As Y is a finite subset, the quotient D 2π must be rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let m n be the representation of D 2π in lowest terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then the orbit of y under the rotation RD forms the vertex set of an inscribed regular n-gon Y ′ ⊆ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As Y does not contain any antipodal pairs, n is necessarily odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Therefore, Y contains the vertex set of a odd regular n-gon Y ′ of the same diameter as Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then Y must coincide with Y ′ as adding any additional point to the set Y ′ would strictly increase the diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The pyramid construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In this section, we describe a class of pointwise extremal subsets of Sn called pyramids in [Kat89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For any pointwise extremal subset Y ⊂ Sn−1, the pyramid construction provides a corresponding pointwise extremal subset in Sn that consists of a rescaled copy of Y together with one extra point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Sn ⊆ Rn+1 be the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Z = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , 0, 1) denote the “north pole”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let xn+1 be the last coordinate of Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then for each plane {xn+1 = a}, a ∈ R that meets Sn at more than one point, the intersection is a rescaled copy of Sn−1 which we call a horizontal section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Each horizontal section contains a suitable rescaled copy of Y which is isometrically embedded into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The pyramid over Y is the subset of Sn consisting of the north pole Z together with a rescaled copy Y ′ of Y inside some horizontal section such that the diameter of Y ′ equals the distance from Z to the horizontal section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Denote by Pyr(Y ) the pyramid over a pointwise extremal subset Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let x, y ∈ Y be points with dSn−1(x, y) = diam(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let x′, y′ ∈ Pyr(Y ) be points corre- sponding to x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then the triple Z, x′, y′ is the vertex set of a spherical equilateral triangle, with spherical angle ∢ x′Zy′ = diam(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Applying the spherical theorem of cosines to the geodesic triangle △ x′Zy′, we obtain the following relationship: diam(Pyr(Y )) = arcsec � sec � diam(Y ) � − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 5 Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7 (The Ak family in S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We apply the pyramid construction to the regular (2k + 1)-gon on S1 to obtain a pointwise extremal configuration Ak ⊆ S2, consisting of the north pole of S2 together with a suitably rescaled copy of the regular (2k + 1)-gon, so that diam(Ak) = arcsec � sec � 2kπ 2k+1 � − 1 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In particular, the diameter diam(Ak) tends to 2π 3 as k goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The configuration A2 consists of the north pole and the vertices of a regular pentagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Construction of k-stacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Following [Kat89], let a β-digon be the convex region on S2 bounded by two meridians (great semicircles joining the north and south poles), with angle β between the two meridians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Given a β-digon, we now introduce a procedure that will be used to produce a certain type of pointwise extremal set Y ⊆ Sn called a k-stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The digon procedure is a “walking process” on the digon that takes as input an odd integer 2k + 1 ≥ 3 and outputs a suitable step length d1 > β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We start walking with equal steps from the north pole on alternating sides of the digon, with step length d1 calibrated so as to get exactly to the south pole after 2k + 1 steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Z ∈ Sn be the north pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A regular n-simplex inscribed in the equator Sn−1 ⊂ Sn defines n + 1 meridians passing through the vertices of the simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let ℓ ∈ (0, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The set of points on Sn which are at distance ℓ away from the north pole Z is a rescaled (n−1)-sphere Sn−1 ℓ , namely a horizontal section of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The intersection between Sn−1 ℓ and the set of n + 1 meridians is the vertex set of an inscribed n-simplex in Sn−1 ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A k-stacked configuration Y (see Figure 2) consists of the north pole Z together with the union of the vertex sets of k stacked n-simplices each obtained as the intersection of a horizontal (n − 1)-sphere Sn−1 ℓi with the n + 1 meridians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The distances ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , ℓk between the horizontal sections and the north pole are determined by the digon procedure as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let d1 be the step length that comes from the digon procedure with input 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Consider the sequence of numbers {dj}2k+1 j=0 where dj is the distance to the north pole of the point obtained after walking j steps in via the digon procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then, the sequence of numbers {ℓi}1≤i≤k is defined in terms of {di}2k+1 i=0 by setting ℓi = d2i for 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Note that d2k = diam(Y ) and d1 = π − d2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Given an odd integer 2k + 1 ≥ 3, the following system of equations summarizes the computation of di for 1 ≤ i ≤ 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 6 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Each value di in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1) is the distance between the point pi shown in this figure and the north pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For 1 ≤ i ≤ 2k + 1, the distance between pi and pi+1 is d1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The two conditions in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1) are obtained by requiring p0 to be the north pole and p2k+1 to be the south pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The conditions in the second line of Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1) are obtained by applying the theorem of cosines for the geodesic spherical triangles with vertices {Z, pi, pi+1}, for each 1 ≤ i ≤ 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The third line Equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1) is obtained by symmetry considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let βn = arccos( 1 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The values {di}0≤i≤2k+1 are determined by n and k via the following equations (see Figure 2): (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1) � � � � � d0 = 0, d2k+1 = π cos(di) cos(di+1) + sin(di) sin(di+1) cos(βn) = cos(d1), 1 ≤ i ≤ 2k di + d2k+1−i = π, 0 ≤ i ≤ 2k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let d1 be the output of the digon procedure with input 2k + 1 on a digon of angle β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If we perform the “walking process” on a digon of angle π − β with complementary step length π − d1, we will eventually get close to the south pole (but will not reach it) and then will start walking back to the north pole and reach it after 2k + 1 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If we add an edge between the points that we traveled during the “walking process”, we obtain the diameter graph (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7) of a regular 2k + 1-gon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='9 (The Bk family in S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When n = 2, for each k, we denote the stacks that result from the digon procedure by Bk, which consists of the vertices of k stacked triangles (2-simplices) together with the north pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Note that B1 coincides with the configuration A1 from Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By construction, diam(Bk) = π − d1 < π − arccos( 1 2) = 2π 3 and limk→∞ diam(Bk) = 2π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The configuration B2 that consists of the north pole and vertices of two stacked triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The green dash lines are meridians;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' the red dot is the north pole, and points of the same color are of the same distance to the north pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10 (The Tk family in S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For each k, we denote the stacks that result from the digon procedure by Tk, which consists of the vertices of k stacked tetrahedra together with the north pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Minimal sets on S2 with diameter below the first accumulation critical value Let d > 0 and let D(S2, d) be the set of all finite subsets Y ⊂ S2 with diam(Y ) < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As each finite subset on S2 is closed, the Hausdorff distance dH is a metric on D(S2, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1 (Diameter-extremal sets in D(S2, d) [Kat89]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A subset Y ∈ D(S2, d) is called diameter-extremal for the diameter functional if there is a little-o function such that diam(Y ) ≤ diam(Y ′) + o( dH(Y, Y ′)) for all Y ′ ⊂ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In other words, we have lim dH(Y ′,Y )→0 diam(Y ′) − diam(Y ) dH(Y ′, Y ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' An n-point set Y is diameter-extremal if and only if at the corresponding point in the configuration space (S2)×n, the gradients of the distances between pairs of points at maximal distance form a taut set (see further in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3 ([Kat89, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A diameter-extremal set Y ∈ D(S2, 2π 3 ) is necessarily pointwise extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4 (Minimal set in D(S2, d) [Kat89]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A subset Y ∈ D(S2, d) is called a minimal set if there is some δ > 0 such that diam(Y ) ≤ diam(Y ′) for all finite subsets Y ′ with dH(Y, Y ′) ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Clearly, every minimal set is diameter-extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In fact, there is a converse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5 ( [Kat89, Theorem 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Every diameter-extremal set in D(S2, 2π 3 ) is a minimal set on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By a mountain-pass argument, one obtains the following consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='6 ([Kat89, Corollary 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' There is exactly one (up to congruence) minimal set in each connected component of D(S2, 2π 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 8 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Configuration space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We will now estimate the number of such connected compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We use the notation k� diam≤d S2 to denote the set of all tuples (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , yk) in �k S2 such that the diameter of its associated set {y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , yk} is less than or equal to d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Note that, for any ϵ > 0, we have a natural continuous map k � diam≤d S2 −→ D(S2, d + ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By realizing k� diam≤d S2 as a closed semi-algebraic set, we obtain the following upper bound on the number of connected components in k� diam≤d S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We set sk = 2k+ k(k+1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then, for every d > 0, the number b0(k, d) of connected components of k� diam≤d S2 satisfies b0(k, d) ≤ 2sk(4sk − 1)3k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We will first describe the set k� diam≤d S2 as a closed basic semi-algebraic set in R3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let xi,j, where 1 ≤ i ≤ k and 1 ≤ j ≤ 3, denote the standard coordinates in R3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then the set k� diam≤d S2 is characterized by the following conditions: � x2 i,1 + x2 i,2 + x2 i,3 = 1 for all 1 ≤ i ≤ k, (xi,1 − xi′,1)2 + (xi,2 − xi′,2)2 + (xi,3 − xi′,3)2 ≤ d2 for all 1 ≤ i < i′ ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Therefore the set k� diam≤d S2 is a basic semi-algebraic set given by sk = 2k + k(k+1) 2 non-strict inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5 implies that b0(k, d) ≤ 1 2(2sk + 2)(2sk + 1)3k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Finiteness results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3 in [Kat89] imply the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='8 ([Kat89]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let 0 < d < 2π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ∈ D(S2, d) be a pointwise extremal set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then for any pair of distinct points y, y′ in Y , the distance dS2(y, y′) is at least arccos � 2 cos2(d) cos2(d/2) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By a packing argument on the sphere, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For each ϵ > 0, there is a positive integer N(ϵ) such that every pointwise extremal subset Y of diameter less than 2π 3 − ϵ contains fewer than N(ϵ) points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For each 0 < ϵ < 2π 3 , there are only finitely many diameter-extremal sets in D(S2, 2π 3 − ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In particular, 2π 3 is the first accumulation point of the critical values of the diameter functional of S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let dϵ = 2π 3 − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5, it suffices to show that there are only finitely many minimal sets in D(S2, dϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='9, there is some N such that every pointwise extremal set in D(S2, dϵ) contains no more than N points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Therefore the image of the continuous map φ φ : N � diam≤dϵ S2 −→ D(S2, 2π 3 ) contains all pointwise extremal configurations with diameter less than or equal to dϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3, the image of φ (in particular) contains all minimal sets with diameter not exceeding dϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Cϵ be the number of connected components which contain a minimal set with diameter no more than dϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='6, the number of minimal sets in D(S2, dϵ) is at most Cϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As the image of φ contains all minimal sets with diameter no more than dϵ, the number Cϵ is bounded by the rank of the map φ∗ : H0 � N � diam≤dϵ S2 � −→ H0 � D(S2, 2π 3 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The claim now follows by invoking the upper bound on the dimension of H0 � N� diam≤dϵ S2 � from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A labeling strategy for the points in Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Recall that Bk ⊆ S2 consists of the north pole and the vertices of k stacked triangles, and that the vertices of the stacked triangles are distributed along three meridians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We label the north pole as Z, then label the vertices of the i-th triangle (counting from the north pole) by Pi, Qi, Ri in such a way that all the Pi, 1 ≤ i ≤ k are on a common longitude and similarly for all Qi, 1 ≤ i ≤ k and Ri, 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The subset dEBk is obtained by removing the points with indexes in E from Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A set Y ⊆ S2 is separable if for each pair of points x, y ∈ Y there are two other points z, w ∈ Y such that the 4-tuple {x, y, z, w} is taut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='13 ([Kat89, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A pointwise extremal subset Y ⊂ S2 with diam(Y ) < 2π 3 is necessarily separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The proof of the above lemma in [Kat89] gives the following stronger result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ⊂ S2 be a subset with diam(Y ) < 2π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Suppose x ∈ Y is held by Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then for any other point y ∈ Y , there exist z, w ∈ Y such that the four-point set {x, y, z, w} is taut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We will now analyze variations of subsets which are continuous with respect to the Haus- dorff distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let {Yt, t ∈ [0, 1]} be a continuous family of subsets of S2 with at most 4 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Suppose the following two conditions hold: the set Y0 is taut, Yt ∈ D(S2, 2π 3 ) for every t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 10 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG Then Yt is taut for each t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As the set Y0 is taut and diam(Y ) < 2π 3 , Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3 implies that the convex hull H0 of Y0 is a tetrahedron and that the origin 0 is in interior of H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For each t ∈ [0, 1] let Ht be the convex hull of the set Yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' To show that each set Yt is taut, it suffices to show that the origin 0 stays in the interior of Ht for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Suppose the contrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let t0 be the supremum of t such that 0 is in the interior of Ht for all smaller values of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Either Ht0 is nondegenerate and then 0 must belong to one of its (triangular) faces, or it is degenerate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=', lies in a plane through the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In either case, we obtain a taut subset of the circle given by the intersection of the plane with the sphere, and can apply Jung’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Namely, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2 we obtain diam(Yt0) ≥ 2π 3 , contradicting the hypothesis Yt0 ∈ D(S2, 2π 3 ) and proving the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Yt, t ∈ [0, 1] be a path in D(S2, 2π 3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If a certain 4-tuple in Y0 is taut, then it continues to be taut for all t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' There exist infinitely many (up to congruence) pointwise extremal sets in D(S2, 2π 3 ) that are not contained in the family Ak or Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Since each connected component contains a (unique) minimal set, it suffices to show that for each k, the configuration dPkBk is separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='14, we can separate most pairs of points from dPkBk except for a pair of points from the triple of points at maximal distance from Pk, namely the points Z, Q1, and R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let us check that such pairs don’t coalesce, either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' This is immediate from the fact that if we remove all layers except the first and the k-th, the remaining configuration is in the connected component in D(S2, 2π 3 ) of the 7-point minimal set B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Thus, by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='16, it suffices to check that if we remove P2 from B2, no remaining points coalesce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' This can be checked directly, and also follows from the fact that the diameter flow applied to the 6-point configuration dP2B2 produces the 6-point minimal set A2 (see Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Anti-self-polar polytopes In this paper, we adopt the following restricted definition of a polytope: a (convex) polytope will be the convex hull of any finite set of points in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The affine hull aff(S) of a set S ⊆ Rn is aff(S) = � k � i=1 αixi ����� k > 0, xi ∈ S, αi ∈ R, k � i=1 αi = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We now give the formal definition of a face of a polytope following [Zie12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2 ([Zie12, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let P ⊆ Rd be a convex polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A linear inequality ⟨c, x⟩ ≤ c0 is valid for P if it is satisfied for all points x ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A face of P is any set of the form F = P ∩ � x ∈ Rd : ⟨c, x⟩ = c0 � where ⟨c, x⟩ ≤ c0 is a valid inequality for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 11 The dimension of a polytope P is defined to be the dimension of its affine hull aff(P) (regarded as an affine space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A 3-dimensional polytope is a polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The codimension- one faces of a polytope P are called facets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' the codimension-two faces are called ridges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If each face of P is a simplex, then P is called a simplicial polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We will use fi(P) to denote the number of i-faces of the polytope P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When there is no risk of confusion, we will denote fi(P) by just fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For a n-dimensional polytope, the vector (f0, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , fn−1) is called the f-vector of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' ASP polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In [Lov83], Lov´asz introduced the following type of polytopes which we will refer to as anti-self-polar (ASP) polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1 Our terminology will be justified in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3 (Anti-self-polar polytopes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let P ⊆ Rn be a n-dimensional polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We say that P is anti-self-polar (ASP) if the following three conditions hold: (1) P is inscribed in the unit sphere Sn−1 ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' (2) P is circumscribed around a sphere centered at the origin with radius s for some 0 < s < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' (3) There is a bijection σ between vertices and facets of P such that if v is any vertex then the facet σ(v) is orthogonal to the vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let P ⊂ Rn be a polytope containing the origin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Sn−1 r (0) be the sphere centered at 0 ∈ Rn with radius r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The polar body of P with respect to the sphere Sn−1 r (0) is defined to be the set polarr(P) = {x ∈ Rn| ⟨x, y⟩ ≤ r2 for all y ∈ P}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As shown in [Hor21], the condition for an ASP polytope in Rn can be restated using the terminology of polar bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In terms of our definition of polarity, if P is an ASP polytope, then there exists some r such that the following relation holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' see [Hor21, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' polarr(P) = −P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The polar body description shows that for each 0 ≤ i ≤ n − 1, the bijection σ in condition (3) can be extended to a bijection between the set of i-dimensional faces and the set of (n − i − 1)-dimensional faces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' see [Hor21, Lemma 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5 ([Kat89, Remark after Theorem 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ⊂ S2 be a pointwise extremal subset with diam(Y ) < 2π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then the convex hull of Y is an ASP polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The result above no longer holds if the restriction on the diameter is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A counterexample is given by an 8-point configuration Y ⊆ S2 consisting of the vertices of an antiprism over a square (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If the diameter of Y is exactly attained by the diagonals of the two squares and by the pairs that consist of a vertex of one square and one of the two farthest vertices of the other square, then Y is pointwise extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' However, the convex hull of Y is not ASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Indeed, note that the top square is a facet of the convex hull of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If the convex hull of Y were ASP, then there would be a vertex y0 ∈ Y such that the distance from y0 to each vertex of the top square would equal diam(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' But, our construction of Y does not satisfy this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 1Lov´asz [Lov83] and Horv`ath[Hor21] use the terminology “strongly self-dual polytopes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 12 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The antiprism on a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ⊆ Rn be a finite subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The diameter graph G(Y ) of Y is defined to be the graph with vertex set V (G) = Y and two vertices y, y′ in G are connected if and only if y and y′ are comaximal in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Given a polytope P, we will refer to the diameter graph of the vertex set of P simply as the diameter graph of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We denote the diameter graph of P by G(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The chromatic number χ(G) of a graph G is the smallest number of colors needed to color the vertices so that no two adjacent vertices share the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The following property of the diameter graph G(P) of an ASP polytope P follows from [Lov83, Lemma 2 and Lemma 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Recall that σ denotes the bijection between the vertex set and the set of facets of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In [Lov83, Lemma 1], it is shown that for any two vertices v, v′ of P, the condition v ∈ σ(v′) is equivalent to v′ ∈ σ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let P be an ASP polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Two vertices v, v′ in G(P) are connected by an edge in G(P) if and only if v ∈ σ(v′), when viewed as vertices in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10 ([Lov83, Theorem 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The diameter graph G(P) of an n-dimensional ASP polytope P ⊆ Rn satisfies χ(G(P)) ≥ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The proof of the theorem is discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The chromatic number of a diameter graph G(Y ) of a subset Y ⊂ Rn is closely related to the following conjecture of Borsuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Borsuk’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='11 (Borsuk’s conjecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y be a bounded subset of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then there is a partition of Y into n + 1 sets each of which has a smaller diameter than Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For finite subsets, Borsuk’s conjecture has the following equivalent form in terms of diam- eter graphs: For every finite bounded subset Y ⊆ Rn, the chromatic number of the diam- eter graph G(Y ) of Y is no greater than n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' To see the above equivalence, a partition {Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , Yk} of Y is equivalent to a coloring of Y by requiring that two points are of the same color if and only if they both belong to EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 13 some Yi, 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Therefore, since Y is a finite set, the condition that the diameter of each subset Yi is less than the diameter of Y is equivalent to requiring that the coloring associated to the partition {Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , Yk} has the property that no two adjacent vertices in the diameter graph G(Y ) share the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Borsuk’s conjecture holds when n = 2 (Borsuk [Bor33]) and n = 3 (Perkal [Per47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The general conjecture was disproved by Khan and Kalai [KK93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The lowest dimensional coun- terexample currently known was constructed by Jenrich and Brouwer (and based on a con- struction by Bondarenko) in dimension 64 [JB14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For additional information on the histori- cal developments on the construction of counterexamples to Borsuk’s conjecture, see [Rai13, Section 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let Y ⊂ Sn−1 be a finite subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Given a regular geodesic n + 1-simplex ∆geodesic n+1 , Sn−1 can be partitioned into n + 1 connected parts {X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , Xn+1} where each Xi contains the interior of one of the faces of ∆geodesic n+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Therefore, by coloring points of Y according to which partition set Xi the point belongs to, we obtain a proper coloring of the diameter graph of Y provided that the diameter diam(Y ) diameter of Y is greater than ηn−1, the diameter of a face of ∆geodesic n+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The above coloring strategy was first described in [Lov83, Section 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Though notice that [Lov83] made a mistake in computing the exact value of ηn−1 [Rai12, Rai13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The correct values of ηn−1 first appeared in [San46] and reproduced in the context of ASP polytopes in [Hor21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10 and the fact that Borsuk’s conjecture is true for n = 3, the chromatic number χ(G(P)) of an ASP polyhedron P ⊆ R3 equals 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In Figures 7 and 8, we display 4-colorings of the diameter graphs of all the ASP polyhedra in Tables 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Borsuk’s conjecture is still open for 4 ≤ n ≤ 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10 suggests that ASP polytopes are a natural source of potential counterexamples to Borsuk’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Additionally, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5, pointwise extremal configurations are closely related to ASP polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2, we present some pointwise extremal subsets on S3 obtained through computer experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' However, the pointwise extremal subsets that we have found so far all have chromatic number 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof of Lovasz’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10 was proved in [Lov83] by analyzing the neighborhood complex of the diameter graph of ASP polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='14 (Neighborhood complex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let G be a finite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The neighborhood complex N(G) is the simplicial complex with vertex set V (G) such that a subset A ⊆ V (G) forms a simplex if and only if the points of A have a neighbor in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In [Lov78], Lov´asz shows the following lower bound of the chromatic number of a graph with respect to the connectivity of its neighborhood complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Recall a topological space X is k-connected if its homotopy groups are trivial up to degree k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='15 ([Lov78]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let G be a graph and suppose that N(G) is k-connected (k ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then χ(G) ≥ k + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='16 ([Lov83, Lemma 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let P be an ASP polytope and G(P) be its diameter graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then N(G(P)) is homotopy equivalent to the boundary of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='16, N(G) is homotopy equivalent to the boundary of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 14 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG As P is a (convex) polytope, the boundary of P is homeomorphic to Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Hence N(G) is homotopy equivalent to Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Therefore, N(G) is (n − 2) connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='15, χ(G) ≥ n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ Let d ≥ 2 and n ≥ 1 be integers and let e(d, n) be the maximum possible number of edges in the diameter graph of a subset of Rd with n points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When d = 2, it is shown in [HP34] that e(2, n) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' This fact leads to one proof of Borsuk’s conjecture for finite subsets Y of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' When d = 3, it was conjectured by V´azsonyi that e(3, n) = 2n − 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' see [Erd46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The V´azsonyi’s conjecture was proved independently by Gr¨unbaum [Gr¨u56], Heppes [Hep56] and Straszewicz [Str57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' As mentioned in Heppes [Hep56], V´azsonyi’s conjecture implies that Borsuk’s conjecture is true for finite subsets in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We have already seen in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10 that the diameter graph of an ASP polytope has high chromatic number, suggesting a possible approach to seeking higher-dimensional counterexamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We now introduce a set of enumerative invariants fij(P) of a polytope P which will be used below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Informally, for i < j, fij(P) counts the number of pairs “i-face contained in a j-face” in the polytope P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Precisely, fij(P) := ♯{(φi, φj) | φi is a i-face of P, φj is a j-face of P, and φi ⊆ φj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='} When there is no risk of confusion, we will simply use fij to denote fij(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Thus f01 is the number of pairs “vertex contained in an edge”, namely just twice the number f1 of edges in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let P be an anti-self-polar polytope of dimension d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let e(G(P)) be the number of edges in the graph G(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then f0d(P) = 2e(G(P)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let V be the set of vertices of P and let W be the set of faces in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Recall that σ denotes the bijection between V and the set of facets of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='9, we have 2e(G(P)) = � v∈V f0(σ(v)) = � φd⊂W f0(φd) = f0d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The second equality above follows from the definition of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Every ASP polyhedron P ⊆ R3 satisfies e(G(P)) = 2f0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='17, we have 2e(G(P)) = f02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Furthermore by duality we have f01 = f12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' This enables us to give a possibly generalizable proof as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Note that, each face has as many vertices as edges and therefore f02 = f12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By duality, f12 = f01 which is twice the number of edges, namely 2f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Thus the number of maximal distances is the same as the number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Meanwhile by the formula for the Euler characteristic, for an anti-self-polar polyhedron we have f1 = 2f0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Altogether, we have f02 = f12 = f01 = 2f1 = 2(2f0 − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Thus the number of maximal distances is also 2f0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ In fact, it is shown in [Kat89] that every pointwise extremal set in S2 with diameter less than 2π 3 exhibits the maximum number of possible edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='19 ([Kat89, Theorem 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Suppose Y ⊂ S2 is a pointwise extremal set with N = |Y | and diam(Y ) < 2π 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then the number of edges in the diameter graph G(Y ) equals 2N − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 15 As noted in [Kat89, page 118], the example of the antiprism on a square constructed in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='13 shows that the above result is no longer true if we remove the diameter constraint: the diameter graph of the antiprism on a square has 8 vertices but only 12 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 4-dimensional polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Consider the V´azsonyi’s problem in R4, that is, for a fixed n, determine the maximal possible number of edges e(4, n) amongst the diameter graphs of all possible n point sets in R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let m be a positive integer and let Y := A ∪ B ⊂ S3 be a subset consisting of 2m points constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The set A consists of m points on an arc of length less than π 2 on a great circle whereas the (disjoint) set B consists of m points also on an arc of length less than π 2 on an orthogonal great circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then each pair of points a ∈ A, b ∈ B is comaximal in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Thus e(4, n) is at least quadratic in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' It is shown in [Erd67] that e(4, n) exactly has quadratic growth rate in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For an anti-self-polar polytope P ⊆ R4, we prove the following lower bound on the number of edges in the diameter graph G(P), originally conjectured in [Kat89, Section 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let P ⊆ R4 be a 4-dimensional anti-self-polar polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Then the number of edges e(G(P)) in the diameter graph G(P) is at least 3f0(P) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='17, the assertion is equivalent to the bound f03(P) ≥ 6f0(P) − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For each facet φ of P, let aj φ be the number of j-gons occurring as faces of φ, and let aj denote the total number of j-gons occurring as faces of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Kalai [Kalai94, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3] proved that every 4-dimensional polytope satisfies g2 ≥ 0 or equivalently a4 + 2a5 + · · · ≥ 4f0(P) − f1(P) − 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let φ run through all the facets of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' By Euler’s formula, we have f03(P) = � φ f0(φ) = � φ 2 + f1(φ) − f2(φ) = � φ 2 + 1 2(3a3 φ + 4a4 φ + 5a5 φ + · · · ) − f2(φ) = � φ 2 + 1 2f2(φ) + 1 2(a4 φ + 2a5 φ + · · · ) = 2f3(P) + f2(P) + 1 2 � φ (a4 φ + 2a5 φ + · · · ) = 2f3(P) + f2(P) + (a4 + 2a5 + · · · ) ≥ 2f3(P) + f2(P) + 4f0(P) − f1(P) − 10 = (2f3(P) + 4f0(P)) + (f2(P) − f1(P)) − 10 = 6f0(P) − 10 by duality, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' □ The above results suggest formulating the following conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 16 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Every ASP polytope P ⊆ R4 satisfies e(G(P)) = 3f0(P) − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2, we report 65 configurations that we generate through numerical experi- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Each of those configurations confirms the above conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Every subset X ⊆ S3 with diam(X) > π 2 satisfies e(G(X)) ≤ 3|X| − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Assuming these conjectures and by an argument similar to the case of the S2 discussed on page 14, one can show that the chromatic number of the diameter graph of any set X in S3 with its diameter greater than π 2 would be at most 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Indeed, Conjecture 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='23 implies that one can always choose a point x0 ∈ X comaximal with at most 5 other points, by the pigeonhole principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Thus, if X − {x0} can be colored with 6 colors, then X can be so colored, also, by using the color not used up by any of its 5 (or fewer) comaximal points, and we conclude by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The fact that this calculation produces the number 6 instead of 5 would provide weak evidence toward the possibility that the Borsuk number of R4 might be the former rather than the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Implementation of the diameter gradient flow This section describes the implementation of the diameter gradient flow on spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Given a finite subset Y of Sn, we first test whether every point in Y is held.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If there is a point y that is not held by Y , we then move y in the direction that points toward the center of the minimum bounding sphere of the tangent vectors determined by points in comaxY (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We continue this process until every point in Y is held.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In other words the point y is updated to a point yt = y0+tv0 ||y0+tv0|| where t > 0 is a parameter value determined through the Armijo rule [Arm66], and v0 is the unit tangent vector at y that points toward the center of the minimum EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 17 bounding sphere of the set {˙γy,y′ | y′ ∈ comaxyY }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The pseudocode of the algorithm is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Algorithm 1: DiameterGradientFlow Input: An initial finite subset Y on unit sphere Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Parameters: β, η ∈ (0, 1) for determing Armijo Rule stepsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Output: The extremal configurations obtained under the diameter gradient flow with initial condition Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 1 Function IsHeld(y, Y ): 2 E ← comaxY (y) 3 Ty(E) ← {˙γy,y′ for y′ ∈ comaxY (y)} 4 if 0 in the convex hull of Ty(E) then 5 return True 6 else 7 return False 8 end if 9 10 Function Main(Y , β, η): 11 /* Initialize convergence tag / 12 tag = False 13 while tag == False do 14 for y0 ∈ Y do 15 if IsHeld(y0, Y ) then 16 tag == True 17 else 18 E ← comaxY (y) 19 Ty0(E) ← {˙γy,y′} for y′ ∈ E} 20 v0 ← center of the minimum bounding sphere of Ty(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 21 /* Determine the step size tk > 0 using Armijo Rule / 22 tk = maxl∈N0 βl s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' diam � Y \\{y0} ∪ � y0+tkv0 ||y0+tkv0|| �� ≤ diam(Y ) − βlη 23 Y ← Y \\{y0} ∪ � y0+tkv0 ||y0+tkv0|| � 24 tag == False 25 break 26 end if 27 end for 28 end while 29 return asee [Arm66] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Computational results In this section, we describe our computational results regarding pointwise extremal config- urations on S2 and S3 using the Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In most of our experiments, we set parameters β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='001 and use the Python package MINIBALL([Dev21]) for finding the optimal direction for decreasing the diameter by moving a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 18 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Pointwise extremal configurations on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In this section, we present the compu- tational results from running the diameter gradient flow Algorithm 1 with initial sets, which are obtained by removing up to six points from Bk (Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='9) with k ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In total, we obtain 54 configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We present in Table 1 the configurations with up to 10 points2 that we found upon convergence of the gradient flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Shape v f r t Diameter Symmetry Group Initial Set A1(= B1) 4 3 4 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='91064 S3 dZB2 A2 6 5 1 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='03446 D5 dP2B2 B2 7 4 3 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='07654 S3 dZB3 C1 8 5 1 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='08707 Z2 d{Q1,P3}B3 A3 8 7 1 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='06459 D7 d{P1,P3}B3 C2 9 5 1 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09335 Z2 d{P1,R1,Q3,Q4}B4 C3 9 5 1 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09079 Z2 dP3B3 C4 9 6 1 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09016 Z2 dP1B3 B3 10 4 6 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09303 S3 dZB4 D1 10 5 1 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09409 {e} d{P1,Q1,P2,Q4,R4,R5}B5 C5 10 5 1 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09317 Z2 d{P1,R3,Q4}B4 C6 10 5 2 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09356 Z2 d{P1,Q1}B4 C7 10 5 3 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09240 Z2 d{P1,R3,P4}B4 D2 10 6 1 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09360 {e} d{P1,R3,R4}B4 C8 10 7 1 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='09174 Z2 d{P1,P3,Q4}B4 A4 10 9 1 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='07654 D9 d{P1,P3,P4}B4 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Pointwise extremal configurations on S2 with up to v = 10 vertices, sorted first by v, then by f (maximal number of edges in a face), then by r (number of faces with a maximal number of edges), then by t (number of triangles in the configuration’s diameter graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' For each of the 10 pointwise extremal configurations that we found, in the last column we list one initial set which leads to that configuration under the diameter gradient flow (a given pointwise extremal configuration may be reached from different initial sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 2An interactive visualization of the table can be found through the link: https://ndag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='io/ anti-self-dual-polyhedra/ EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 19 (a) C1 (b) C2 (c) C3 (d) C4 (e) C5 (f) C6 (g) C7 (h) C8 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Eight Z2 symmetric pointwise extremal configurations with at most 10 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' (a) D1 (b) D2 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Two asymmetric pointwise extremal configurations with 10 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 20 MIKHAIL KATZ, FACUNDO M´EMOLI, AND QINGSONG WANG (a) C1 (b) C2 (c) C3 (d) C4 (e) C5 (f) C6 (g) C7 (h) C8 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Diameter graphs of Z2 symmetric pointwise extremal configura- tions with less than 10 with a minimal coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Note that all diameter graphs above can be colored with four colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' (a) D1 (b) D2 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The diameter graph of the two asymmetric pointwise extremal configurations D1 and D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Pointwise extremal configurations on S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' In this section we present some compu- tational results on pointwise extremal configurations on S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Recall that Tk ⊆ S3 denotes the k-stack;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' The Tk consists of the north pole and the vertices of k stacked 3-simplices, for a total of 4k + 1 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We use similar indexing for the points in Tk, that is, the north pole is denoted Z, then we label the verticees of i-th tetrahedron (counting from the north pole) by Pi, Qi, Ri, Si in such a way that all the Pi, 1 ≤ i ≤ k are on a common longitude and similarly for all Qi, 1 ≤ i ≤ k, Ri, 1 ≤ i ≤ k, and Si, 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Applying the diameter gradient flow to the initial sets of the diameter gradient flow be the subsets of T1, T2, T3, T4 with at most four points removed, one obtains at least 65 distinct pointwise-extremal configurations which are not pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 3 3A comprehensive table containing statistics for the 65 configurations, similar to Table 1, can be accessed through the following link: https://ndag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='io/anti-self-dual-polyhedra/table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='html EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 21 Through exact calculation via the Python package NetworkX([HSS]), we find that the diameter graph of each of these 65 configurations has chromatic number equal to 5 and also satisfies e = 3v − 5 where e and v are the number of edges and the number of vertices in the diameter graph, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Semi-algebraic sets Let k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let R[x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , xk] be the k-dimensional ring of polynomials with real coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' We now introduce the notion of semi-algebraic subset following [BCR13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1 ([BCR13, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Let {ri}s i=1 be a set of positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A semi-algebraic subset of Rn is a subset of the form s� i=1 ri � j=1 {x ∈ Rn | fi,j ∗i,j 0} , where fi,j ∈ R [X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , Xn] and the operation ∗i,j is either < or =, for i = 1, , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , s and j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A collection A of subsets of a set X is called an algebra of sets if A contains the empty set and is closed under finite union, finite intersection and under taking complements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Semi-algebraic subsets of Rn form the smallest algebra of sets that contains all sets of the form {x ∈ Rn | f(x) > 0} , where f ∈ R [X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , Xn] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='4 ([BCR13, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A basic open semi-algebraic subset of Rn is a set of the form {x ∈ Rn | f1(x) > 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , fs(x) > 0} where f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , fs ∈ R [X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , Xn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' A basic closed semi-algebraic subset of Rn is a set of the form {x ∈ Rn | f1(x) ≥ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , fs(x) ≥ 0} where f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , fs ∈ R [X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , Xn] By applying Morse theory, Milnor [Mil64] obtained the following bound on the number of Betti numbers of a closed basic semi-algebraic set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='5 ([Mil64, Theorem 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' If X ⊂ Rn is a basic closed semi-algebraic subset defined by p polynomial inequalities f1 ≥ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' , fp ≥ 0 of degree ≤ d, then the sum of the Betti numbers of X is at most 1 2(dp + 2)(dp + 1)n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' EXTREMAL SPHERICAL POLYTOPES AND BORSUK’S CONJECTURE 23 [Wal70] David W Walkup, The lower bound conjecture for 3-and 4-manifolds, Acta Mathematica 125 (1970), 75–107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' [Zie12] G¨unter M Ziegler, Lectures on polytopes, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' 152, Springer Science & Business Media, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Bar Ilan University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Email address: katzmik@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='biu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='il The Ohio State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Email address: facundo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='memoli@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='com University of Utah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content=' Email address: qswang@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='utah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FPT4oBgHgl3EQfZDSN/content/2301.13076v1.pdf'}