On the integer points in a lattice polytope: n-fold Minkowski sum and boundary

Abstract

In this article we compare the set of integer points in the homothetic copy $n\Pi$ of a lattice polytope $\Pi \subseteq {\mathbb{R}}^d$ with the set of all sums $x_1+\cdots +x_n$ with $x_1,...,x_n\in \Pi \cap {\mathbb{Z}}^d$ and $n\in \mathbb{N}$. We give conditions on the polytope $\Pi$ under which these two sets coincide and we discuss two notions of boundary for subsets of ${\mathbb{Z}}^d$ or, more generally, subsets of a finitely generated discrete group.