On the integer points in a lattice polytope: \(n\)-fold Minkowski sum and boundary (Q717837)

In this article the authors 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 \(x_1,\dots,x_n\in \Pi\cap \mathbb{Z}^d\) and \(n\in \mathbb N\). They prove that if a polytope \(\Pi\) possess a triangulation into lattice triangles of lattice volume 1 (or, equivalently, of Euclidean volume \(1/(d!)\)) then the above two sets coincide. Further the authors discuss two notions of boundary for subsets of \(\mathbb Z^d\) or, more generally, subsets of a finitely generated discrete group.