ISSN 0032-9460, Problems of Information Transmission, 2012, Vol. 48, No. 2, pp. 185–192.
Pleiades Publishing, Inc., 2012.
Original Russian Text
K.Yu. Gorbunov, A.V. Seliverstov, V.A. Lyubetsky, 2012, published in Problemy Peredachi Informatsii, 2012, Vol. 48, No. 2,
Geometric Relationship between Parallel Hyperplanes,
Quadrics, and Vertices of a Hypercube
K. Yu. Gorbunov, A. V. Seliverstov, and V. A. Lyubetsky
Kharkevich Institute for Information Transmission Problems, RAS, Moscow
firstname.lastname@example.org email@example.com firstname.lastname@example.org
Received November 15, 2011; in ﬁnal form, January 23, 2012
Abstract—In a space of dimension 30 we ﬁnd a pair of parallel hyperplanes, uniquely deter-
mined 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.
1. PROBLEM SETTING
Our results are valid for any linearly ordered ﬁeld, but for brevity we will speak about the real
ﬁeld R.Ahypercube (or simply a cube) is a polyhedron in a space of dimension n whose vertices
have coordinates 0 or 1. The weight of a vertex of a cube is the number of its unit coordinates.
A quadric in R
is the set of zeros of a real (possibly, reducible) quadratic polynomial f (x
A quadric f = 0 is said to be empty if values of the polynomial f at vertices of the cube are either
all nonnegative or all nonpositive. In what follows, we assume the ﬁrst case. We consider the
structure of the set of minimum points of a quadratic polynomial f on the set of all vertices of the
cube, i.e., location of an empty quadric with respect to the cube. A particular case of a quadric is a
pair of parallel hyperplanes. In more detail, we consider the possibility of special arrangements of
parallel hyperplanes with respect to vertices of the cube or to an arbitrary set of points in general
position in an n-space.
Finding the minimum of a general quadratic polynomial on the set of vertices of the cube is an
algorithmically hard problem. Eﬃcient algorithms—for instance, the pseudo-Boolean programming
method —are applicable in particular cases only. In [2, 3], an overview of heuristic algorithms
is presented. A particular case of optimization problems is the case of problems with separated
variables. In  (see also references therein), a multicriteria minimax problem is considered where
optimization of quadratic forms is performed over sets of vertices of two unit cubes of diﬀerent
dimensions. Results of  imply that vertices of the cube lying on an empty quadric lie on a empty
cylinder (non-full-rank quadric) that does not contain other vertices. A degenerate case consists in
describing vertices of the cube that lie on a pair of coinciding hyperplanes.
Many papers (see, e.g., [6–9]) give estimates for the number of vertices lying on a quadric.
A hyperplane contains at most half of all vertices of the cube. However, if the corresponding linear
function depends on each of the n variables nontrivially, the fraction of vertices of the cube lying
on this hyperplane tends to zero with growing n . A similar result was obtained for a quadratic
polynomial with suﬃciently many monomials: the fraction of vertices of the cube at which such a