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arXiv · 1207.0164

Lattice-point generating functions for free sums of convex sets

Matthias Beck·Pallavi Jayawant·Tyrrell B. McAllister
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Abstract

Let $\J$ and $\K$ be convex sets in $\R^{n}$ whose affine spans intersect at a single rational point in $\J \cap \K$, and let $\J \oplus \K = \conv(\J \cup \K)$. We give formulas for the generating function {equation*} \sigma_{\cone(\J \oplus \K)}(z_1,..., z_n, z_{n+1}) = \sum_{(m_1,..., m_n) \in t(\J \oplus \K) \cap \Z^{n}} z_1^{m_1}... z_n^{m_n} z_{n+1}^{t} {equation*} of lattice points in all integer dilates of $\J \oplus \K$ in terms of $\sigma_{\cone \J}$ and $\sigma_{\cone \K}$, under various conditions on $\J$ and $\K$. This work is motivated by (and recovers) a product formula of B.\ Braun for the Ehrhart series of $\P \oplus \Q$ in the case where $\P$ and $\Q$ are lattice polytopes containing the origin, one of which is reflexive. In particular, we find necessary and sufficient conditions for Braun's formula and its multivariate analogue.

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Matthias Beck, Pallavi Jayawant, Tyrrell B. McAllister. 2012-06-30. Lattice-point generating functions for free sums of convex sets. https://arxiv.org/abs/1207.0164

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