Polyhedra with the Integer Carathéodory Property

A polyhedron P has the Integer Carathéodory Property if the following holds. For any positive integer k and any integer vector w∈kP, there exist affinely independent integer vectors x₁,...,x_t∈P and positive integers n₁,...,n_t such that n₁+...+n_t=k and w=n1x1+...+ntxt. In this paper we prove that if P is a (poly)matroid base polytope or if P is defined by a totally unimodular matrix, then P and projections of P have the Integer Carathéodory Property. For the matroid base polytope this answers a question by Cunningham from 1984.