On the integer points in a lattice polytope: n-fold Minkowski sum and boundary
This is a theoretical contribution to discrete geometry and integer programming, providing conditions for when these sets coincide.
The paper compares integer points in the n-fold Minkowski sum of a lattice polytope with sums of n integer points in the original polytope, giving conditions for their equality and discussing boundary notions for discrete groups.
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.