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 x1 + . . . + xn with \({x_1,\ldots,x_n\in \Pi\cap{{\mathbb Z}}^d}\) 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.