The cardinality of the separated vertex set of a multidimensional cube

An $n$-dimensinal cube and a sphere inscribed into it are considered. The conjecture of A. Ben-Tal, A. Nemirovskii, C. Roos states that each tangent hyperplane to the sphere strictly separates not more than $2^{n-2}$ cube vertices. In this paper this conjecture is proved for $n\leq 6.$ New examples of hyperplanes separating exactly $2^{n-2}$ cube vertices are constructed for any $n$. It is proved that hyperplanes orthogonal to radius vectors of cube vertices separate less than $2^{n-2}$ cube vertices for $n\ge3$.