Lattice-point generating functions for free sums of convex sets

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.Comment: 17 pages, 2 figures, to appear in Journal of Combinatorial Theory Series