Geometric relationship between parallel hyperplanes, quadrics, and vertices of a hypercube

Geometric relationship between parallel hyperplanes, quadrics, and vertices of a hypercube 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Problems of Information Transmission Springer Journals

Geometric relationship between parallel hyperplanes, quadrics, and vertices of a hypercube

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Publisher
SP MAIK Nauka/Interperiodica
Copyright
Copyright © 2012 by Pleiades Publishing, Ltd.
Subject
Engineering; Information Storage and Retrieval; Systems Theory, Control; Electrical Engineering; Communications Engineering, Networks
ISSN
0032-9460
eISSN
1608-3253
D.O.I.
10.1134/S0032946012020081
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Problems of Information TransmissionSpringer Journals

Published: Jul 13, 2012

References

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