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In this work, the classical Borsuk conjecture is discussed, which states thatany set of diameter 1 in the Euclidean space $ {\mathbb R}^d $ can be dividedinto $ d+1 $ parts of smaller diameter. During the last two decades, manycounterexamples to the conjecture have been proposed in high dimensions.However, all of them are sets of diameter 1 that lie on spheres whose radii areclose to the value $ {1}{\sqrt{2}} $. The main result of this paper is asfollows: {\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 $.

Counterexamples to Borsuk's conjecture on spheres of small radii