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.
Categories: math.CO