Notes on lattice points of zonotopes and lattice-face polytopes

2010/06/29 by Bey, Christian, Henk, Martin, Henze, Matthias +1
#Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1006.5574

Abstract

Minkowski's second theorem on successive minima gives an upper bound on the volume of a convex body in terms of its successive minima. We study the problem to generalize Minkowski's bound by replacing the volume by the lattice point enumerator of a convex body. In this context we are interested in bounds on the coefficients of Ehrhart polynomials of lattice polytopes via the successive minima. Our results for lattice zonotopes and lattice-face polytopes imply, in particular, that for 0-symmetric lattice-face polytopes and lattice parallelepipeds the volume can be replaced by the lattice point enumerator.

Related

• Successive Minima and Lattice Points
• On counterexamples to a conjecture of Wills and Ehrhart polynomials whose roots have equal real parts
• Bounds on the lattice point enumerator via slices and projections
• A note on lattice-face polytopes and their Ehrhart polynomials
• Enumeration of lattice polytopes by their volume