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

TL;DR

This paper investigates the relationship between integer points in scaled lattice polytopes and sums of points within the polytope, providing conditions for their equality and exploring boundary notions in discrete groups.

Contribution

It establishes conditions under which the integer points in a scaled polytope equal the sum of points within the polytope and discusses boundary concepts in discrete groups.

Findings

Identifies conditions for equality of integer points and Minkowski sums in lattice polytopes.

Analyzes boundary notions for subsets of integer lattice points and discrete groups.

Abstract

In this article we compare the set of integer points in the homothetic copy $n\Pi$ of a lattice polytope $\Pi\subseteq\R^d$ with the set of all sums $x_1+\cdots+x_n$ with $x_1,...,x_n\in \Pi\cap\Z^d$ and $n\in\N$. We give conditions on the polytope $\Pi$ 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.