  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.
