Geodesic
The cardinality of the separated vertex set of a multidimensional cube
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2010), pp. 11-17

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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 $n\ge3$. 
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     author = {I. N. Shnurnikov},
     title = {The cardinality of the separated vertex set of a multidimensional cube},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
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     publisher = {mathdoc},
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     url = {http://geodesic.mathdoc.fr/item/VMUMM_2010_2_a1/}
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I. N. Shnurnikov. The cardinality of the separated vertex set of a multidimensional cube. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2010), pp. 11-17. http://geodesic.mathdoc.fr/item/VMUMM_2010_2_a1/