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2010-27

Notes on lattice points of zonotopes and lattice-face polytopes

by Henk, Martin, Henze, Matthias, Linke, Eva.


Series: 2010-27, Preprints

MSC:
52C07 Lattices and convex bodies in $n$ dimensions
52B20 Lattice polytopes (including relations with commutative algebra and algebraic geometry)
52A40 Inequalities and extremum problems
11H06 Lattices and convex bodies

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. To this 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.

Keywords:
Zonotope, lattice-face polytope, Ehrhart polynomial, successive minima