Lattice-point generating functions for free sums of convex sets

Abstract

Let J and K be convex sets in Rn whose affine spans intersect at a single rational point in J∩K, and let J⊕K=conv(J∪K). We give formulas for the generating functionσcone(J⊕K)(z1,…,zn,zn+1)=∑(m1,…,mn)∈t(J⊕K)∩Znz1m1⋯znmnzn+1t of lattice points in all integer dilates of J⊕K in terms of σconeJ and σconeK, 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⊕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 Braunʼs formula and its multivariate analogue.