Counterexamples to Borsuk's conjecture on spheres of small radii

Details Counterexamples to Borsuk's conjecture on spheres of small radii Counterexamples to Borsuk's conjecture on spheres of small radii

2010-10-03

arxiv.org/abs/1010.0383v1

Mathematics

Combinatorics

13 pages

Scientific paper

In this work, the classical Borsuk conjecture is discussed, which states that any set of diameter 1 in the Euclidean space $ {\mathbb R}^d $ can be divided into $ d+1 $ parts of smaller diameter. During the last two decades, many counterexamples to the conjecture have been proposed in high dimensions. However, all of them are sets of diameter 1 that lie on spheres whose radii are close to the value $ {1}{\sqrt{2}} $. The main result of this paper is as follows: {\it for any $ r > {1}{2} $, there exists a $ d_0 $ such that for all $ d \ge d_0 $, a counterexample to Borsuk's conjecture can be found on a sphere $ S_r^{d-1} \subset {\mathbb R}^d $.

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