Geometric relationship between parallel hyperplanes, quadrics, and vertices of a hypercubeстатья
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Дата последнего поиска статьи во внешних источниках: 30 марта 2016 г.
Аннотация:In a space of dimension 30 we find a pair of parallel hyperplanes, uniquely determined by vertices of a unit cube lying on them, such that strictly between the hyperplanes there are no vertices of the cube, though there are integer points. A similar two-sided example is constructed in dimension 37. We consider possible locations of empty quadrics with respect to vertices of the cube, which is a particular case of a discrete optimization problem for a quadratic polynomial on the set of vertices of the cube. We demonstrate existence of a large number of pairs of parallel hyperplanes such that each pair contains a large number of points of a prescribed set.