The cardinality of the separated vertex set of a multidimensional cube |
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Authors: | I N Shnurnikov |
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Institution: | (3) Institute of Mathematics and Informatics, Vilnius, Lithuania; |
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Abstract: | 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 2
n−2 cube vertices. In this paper this conjecture is proved for n ≤ 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 ≥ 3. |
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Keywords: | |
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