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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A Brunn-Minkowski inequality for the integer lattice
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by R. J. Gardner and P. Gronchi PDF
Trans. Amer. Math. Soc. 353 (2001), 3995-4024 Request permission

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
  • MR Author ID: 195745
  • 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
  • MR Author ID: 340283
  • Email: paolo@iaga.fi.cnr.it
  • 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
  • © Copyright 2001 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 3995-4024
  • MSC (1991): Primary 05B50, 52B20, 52C05, 52C07; Secondary 92C55
  • DOI: https://doi.org/10.1090/S0002-9947-01-02763-5
  • MathSciNet review: 1837217