Lattice-point generating functions for free sums of convex sets

Let J and K be convex sets in R n 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 function σ cone ( J ⊕ K ) ( z 1 , … , z n , z n + 1 ) = ∑ ( m 1 , … , m n ) ∈ t ( J ⊕ K ) ∩ Z n z 1 m 1 ⋯ z n m n z n + 1 t of lattice points in all integer dilates of J ⊕ K in terms of σ cone J and σ 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 ⊕ 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.