Summary: Let \(\mathcal J\) and \(\mathcal K\) be convex sets in \(\mathbb R^n\) whose affine spans intersect at a single rational point in \(\mathcal J\cap\mathcal K\), and let \(\mathcal J\oplus\mathcal K=\mathrm{conv}(\mathcal J\cup\mathcal K)\). We give formulas for the generating function \[ \sigma_{\mathrm{cone}(\mathcal J\oplus \mathcal K)}(z_1,\dots,z_n,z_{n+1}) =\sum_{(m_1,\dots,m_n)\in t(\mathcal J\oplus \mathcal K)\cap \mathbb Z^n} z^{m_1}_1\cdots z_n^{m_n}z_{n+1}^t \] of lattice points in all integer dilates of \(\mathcal J\oplus\mathcal K\) in terms of \(\sigma_{\mathrm{cone}\mathcal J}\) and \({\sigma}_{\mathrm{cone}\mathcal K}\), under various conditions on \(\mathcal J\) and \(\mathcal K\).
This work is motivated by (and recovers) a product formula of B. Braun [Electron. J. Comb. 13, No. 1, Research paper N15, 5 p. (2006; Zbl 1109.52011)] for the Ehrhart series of \(\mathcal P\oplus\mathcal Q\) in the case where \(\mathcal P\) and \(\mathcal Q\) 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.