Convex cones, integral zonotopes, limit shape

Given a convex cone C in Rd, an integral zonotope T is the sum of segments [0,vi] (i=1,…,m) where each vi∈C is a vector with integer coordinates. The endpoint of T is k=∑1mvi. Let T(C,k) be the family of all integral zonotopes in C whose endpoint is k∈C. We prove that, for large k, the zonotopes in T(C,k) have a limit shape, meaning that, after suitable scaling, the overwhelming majority of the zonotopes in T(C,k) are very close to a fixed convex set. We also establish several combinatorial properties of a typical zonotope in T(C,k).