A Brunn-Minkowski inequality for the integer lattice

Abstract

Abstract. A close discrete analog of the classical Brunn-Minkowksi inequality that holds for finite subsets of the integer lattice is obtained. This is applied to obtain strong new lower bounds for the cardinality of the sum of two finite sets, one of which has full dimension, and, in fact, a method for computing the exact lower bound in this situation, given the dimension of the lattice and the cardinalities of the two sets. These bounds in turn imply corresponding new bounds for the lattice point enumerator of the Minkowski sum of two convex lattice polytopes. A Rogers-Shephard type inequality for the lattice point enumerator in the plane is also proved.

On Sums of Generating Sets in ℤ2n

“…Naturally, the structural properties of G play a role in this problem as well. For the torsion-free case, G = Z d , this question and similar ones were discussed by Ruzsa [30], and a full answer was finally given by Gardner and Gronchi [15]. In the extremal construction, the smaller set is a simplex of d + 1 points, on one of whose edges lies an arithmetic progression, and the other set is roughly the sum of several copies of it.…”

Majorization and Rényi entropy inequalities via Sperner theory

“…Even for the subclass of uniform distributions and for α = 1, however, one can easily construct counterexamples showing that an exponent of 2/d fails in general in Z d . We remark that in order to develop a discrete Brunn-Minkowski inequality in the integer lattice, Gardner and Gronchi [55] imposed a natural and appropriate dimensional assumption, the main point of which is that at least two points should be assigned to each axis direction. However, the dimensional assumption from [55] is still not sufficient to obtain an improvement of Theorem 2.6 with exponent 1+α d (as can be checked by counterexamples).…”

“…We remark that in order to develop a discrete Brunn-Minkowski inequality in the integer lattice, Gardner and Gronchi [55] imposed a natural and appropriate dimensional assumption, the main point of which is that at least two points should be assigned to each axis direction. However, the dimensional assumption from [55] is still not sufficient to obtain an improvement of Theorem 2.6 with exponent 1+α d (as can be checked by counterexamples). Hence we leave the discovery of appropriate dimensional entropy inequalities in the integer lattice as an open question for future works.…”

Generalizations of a Matrix Inequality

“…The well-known Brunn-Minkowski inequality is one of the most important inequalities in geometry. There are many other interesting results related to the Brunn-Minkowski inequality (see [1][2][3][4][5][6][7][8]). The matrix form of the Brunn-Minkowski inequality (see [9,10]) asserts that if A and B are two positive definite matrices of order n and 0 1 λ < < , then ( ) ( )…”