Document Type
Article
Publication Date
2001
Keywords
Brunn-Minkowski inequality, Lattice, Lattice polygon, Convex lattice polytope, Lattice point enumerator, Sum set, Difference set
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, Paolo, "A Brunn-Minkowski Inequality for the Integer Lattice" (2001). Mathematics Faculty Publications. 27.
https://cedar.wwu.edu/math_facpubs/27
Subjects - Topical (LCSH)
Convex polytopes; Lattice theory
Genre/Form
articles
Type
Text
Rights
Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.
Language
English
Format
application/pdf
