  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 $.
