Integer Decomposition Property of Dilated Polytopes

David A. Cox, Christian Haase, Takayuki Hibi, Akihiro Higashitani


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.


dilated polytope, integer decomposition property, Hilbert basis.

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