Lattice-point generating functions for free sums of convex sets

Let JJ and KK be convex sets in RnRn whose affine spans intersect at a single rational point in J∩KJ∩K, and let J⊕K=conv(J∪K)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⊕KJ⊕K in terms of σconeJσconeJ and σconeKσconeK, under various conditions on JJ and KK. This work is motivated by (and recovers) a product formula of B. Braun for the Ehrhart series of P⊕QP⊕Q in the case where PP and QQ 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.