Integer Decomposition Property of Dilated Polytopes

An integral convex polytope $\mathcal{P} \subset \mathbb{R}^N$ possesses the integer decomposition property if, for any integer $k > 0$ and for any $\alpha \in k \mathcal{P} \cap \mathbb{Z}^{N}$, there exist $\alpha_{1}, \ldots, \alpha_k \in \mathcal{P} \cap \mathbb{Z}^{N}$ 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.