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The Cardinality of the Separated Vertex Set of a Multidimensional Cube / Shnurnikov I.N. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2010. № 2. P. 11-17 [Moscow Univ. Math. Bulletin. Vol. 65, No 2, 2010. P. 63-68]. An n-dimensional cube and the sphere inscribed into it are considered. The conjecture of A. Ben-Tal, A. Nemirovski, and C. Roos states that each tangent hyperplane to the sphere strictly separates not more than 2n-2 cube vertices. In this paper this conjecture is proved for n ≤ 6. New examples of hyperplanes separating exactly 2n-2 cube vertices are constructed for any n. It is proved that hyperplanes orthogonal to radius vectors of cube vertices separate less than 2n-2 cube vertices for n ≥ 3.
Key words: threshold functions, separated vertices of cube.
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