Using The Borsuk-Ulam Theorem

By Jirí Matousek. 2007

840 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9983e5bd7be99e848e5b4f5ab86852d2668b332d

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In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Topology and combinatorics of partitions of masses by hyperplanes (2003, background): “upy one of central positions in this ﬁeld. In this category are R. Rado’s “theorem for general measures” [32], B. Gru¨nbaum’s “center point theorem” for convex bodies [19], etc. Survey articles [15], [29], [46], [47], [48] as well as the original papers [6], [33], [37], [51], cover diﬀerent aspects ”
Optimal bounds for a colorful Tverberg--Vrecica type problem (2009, background): “(See [Mat03] for an introduction to these notions.”
Partitions of nonzero elements of a finite field into pairs (2010, methodology): “bounds for the chromatic number of the Kneser graphs in [17], other examples of topological methods can be found in [18, 24].”
The endpoint case of the Bennett–Carbery–Tao multilinear Kakeya conjecture (2008, background): “For a proof of the Borsuk‐Ulam theorem, the reader may look at Hatcher’s book on algebraic topology [8, pp.174‐176]. Another reference is [ 12 ] by Matouˇsek.”
Topological lower bounds for the chromatic number: A hierarchy (2002, methodology): “omplement of G, and hence NP-complete and hard to approximate; see, e.g., Ausiello et al. [2]. Simplicial complexes. We use letters like K,L,... to denote simplicial complexes. (See, e.g., [29], [5], [25] for more background). We consider only ﬁnite simplicial complexes, so a simplicial complex K is”
Fast construction of nets in low dimensional metrics, and their applications (2004, neutral): “Borsuk-Ulam theorem (cf. [37]) states that there exists x ∈ S k such that φ ′ ( x ) = φ ′ ( x ).”
The chromatic thresholds of graphs (2011, methodology): “It follows from the Borsuk-Ulam Theorem (see [32], for example) that χ (”
Curves of Finite Total Curvature (2006, background): “β. In fact, these arcs can be chosen to depend continuously on v. Now let a(v) ∈S1 be the midpoint of α(v). Consider the continuous map a: S d−1→S1 ⊂E . By one version of the Borsuk/Ulam theorem (see [Mat03]), there must be some v such that a(v) = a(−v). But the height functions for v and −v are neg”