Notes on lattice points of zonotopes and lattice-face polytopes

Understand

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.