Counterexamples to Borsuk's conjecture on spheres of small radii

TL;DR

This paper demonstrates that counterexamples to Borsuk's conjecture exist on spheres with radii greater than 1/2 in sufficiently high dimensions, extending previous results limited to spheres near radius 1/√2.

Contribution

It proves the existence of counterexamples on spheres of radius greater than 1/2 in high dimensions, broadening the scope of known counterexamples to Borsuk's conjecture.

Findings

Counterexamples exist on spheres with radius > 1/2 in high dimensions.

Previous counterexamples were limited to spheres near radius 1/√2.

The result applies to all sufficiently high dimensions.

Abstract

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 $.