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THE BOUNDARY VOLUME OF A LATTICE POLYTOPE

Part of: Polytopes and polyhedra

Published online by Cambridge University Press:  26 September 2011

GÁBOR HEGEDÜS  and
ALEXANDER M. KASPRZYK
GÁBOR HEGEDÜS
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstraße 69, A-4040 Linz, Austria (email: gabor.hegedues@oeaw.ac.at)
ALEXANDER M. KASPRZYK*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom (email: a.m.kasprzyk@imperial.ac.uk)
*
✉For correspondence; e-mail: a.m.kasprzyk@imperial.ac.uk
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Abstract

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For a d-dimensional convex lattice polytope P, a formula for the boundary volume vol(∂P) is derived in terms of the number of boundary lattice points on the first ⌊d/2⌋ dilations of P. As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulas for the f-vector of a smooth polytope in dimensions three, four, and five. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.

Keywords

lattice polytopeboundary volumereflexive polytopeorder polytopeBirkhoff polytope

MSC classification

Secondary: 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) 52B12: Special polytopes (linear programming, centrally symmetric, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 1 , February 2012 , pp. 84 - 104
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

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