Counterexamples to Borsuk's conjecture on spheres of small radii

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 $S_r^{d-1} \subset {\mathbb R}^d$.