Borsuk number of special sets on spheres of small radii

In 1933 K. Borsuk stated his classical conjecture that any set of diameter 1 in the d-dimensional Euclidean space can be divided into d + 1 parts of smaller diameters. In 1993 Borsuk’s conjecture was disproved. Moreover, in 2012 it was proved that counterexamples to the conjecture can be found on spheres of any radii greater than 1/2. In this paper we build using (-1, 1)-vectors and (-1, 0, 1)-vectors new counterexamples to Borsuk’s conjecture on spheres of small radii in Rd.