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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A Brunn-Minkowski inequality for the integer lattice


Authors: R. J. Gardner and P. Gronchi
Journal: Trans. Amer. Math. Soc. 353 (2001), 3995-4024
MSC (1991): Primary 05B50, 52B20, 52C05, 52C07; Secondary 92C55
Published electronically: June 6, 2001
MathSciNet review: 1837217
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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.


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Additional Information

R. J. Gardner
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063
Email: gardner@baker.math.wwu.edu

P. Gronchi
Affiliation: Istituto di Analisi Globale ed Applicazioni, Consiglio Nazionale delle Ricerche, Via S. Marta 13/A, 50139 Firenze, Italy
Email: paolo@iaga.fi.cnr.it

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02763-5
PII: S 0002-9947(01)02763-5
Keywords: Brunn-Minkowski inequality, lattice, lattice polygon, convex lattice polytope, lattice point enumerator, sum set, difference set
Received by editor(s): September 30, 1999
Published electronically: June 6, 2001
Additional Notes: First author supported in part by U.S. National Science Foundation Grant DMS-9802388
Article copyright: © Copyright 2001 American Mathematical Society