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Notes on lattice points of zonotopes and lattice-face polytopes

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

Series: 2010-27, Preprints

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

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.

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