Notes on lattice points of zonotopes and lattice-face polytopes

TL;DR

This paper explores generalizations of Minkowski's second theorem by relating lattice point counts of convex bodies to their successive minima, focusing on lattice zonotopes and face polytopes.

Contribution

It introduces bounds on Ehrhart polynomial coefficients for lattice polytopes using successive minima, extending Minkowski's volume bounds to lattice point enumeration.

Findings

Volume bounds can be replaced by lattice point counts for certain polytopes.

Results apply specifically to 0-symmetric lattice-face polytopes and parallelepipeds.

Provides new inequalities linking geometry and lattice point enumeration.

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.