#### Document Type

Article

#### Publication Date

2001

#### 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.

#### Publication Title

Transactions of the American Mathematical Society

#### Volume

353

#### Issue

10

#### First Page

3995

#### Last Page

4024

#### Required Publisher's Statement

First published in The Transactions of the American Mathematical Society in Volume 353, Number 10, 2001, published by the American Mathematical Society

#### Recommended Citation

Gardner, Richard J. and Gronchi, P., "A Brunn-Minkowski Inequality for the Integer Lattice" (2001). *Mathematics.* Paper 27.

http://cedar.wwu.edu/math_facpubs/27