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 $nΠ$ of a lattice polytope $Π\subseteq\R^d$ with the set of all sums $x_1+\cdots+x_n$ with $x_1,...,x_n\in Π\cap\Z^d$ and $n\in\N$. We give conditions on the polytope $Π$ under which these two sets coincide and we discuss two notions of boundary for subsets of $\Z^d$ or, more generally, subsets of a finitely generated discrete group.