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