Integer decomposition property of dilated polytopes

Abstract

Let $\mathcal{P}\subset {\mathbb{R}}^N$ be an integral convex polytope of dimension $d$ and write $k\mathcal{P}$, where $k=1,2,\dots$, for dilations of $\mathcal{P}$. We say that $\mathcal{P}$ possesses the integer decomposition property if, for any integer $k=1,2,\dots$ and for any $\alpha \in k\mathcal{P}\cap {\mathbb{Z}}^N$, there exist ${\alpha }_1,\dots ,{\alpha }_k$ belonging to $\mathcal{P}\cap {\mathbb{Z}}^N$ such that $\alpha ={\alpha }_1+\cdots +{\alpha }_k$. A fundamental question is to determine the integers $k>0$ for which the dilated polytope $k\mathcal{P}$ possesses the integer decomposition property. In the present paper, combinatorial invariants related to the integer decomposition property of dilated polytopes will be proposed and studied.

16 pages, comments welcome