Polyhedra with the Integer Caratheodory Property

A polyhedron P has the Integer Caratheodory Property if the following holds. For any positive integer k and any integer vector w in kP, there exist affinely independent integer vectors x_1,...,x_t in P and positive integers n_1,...,n_t such that n_1+...+n_t=k and w=n_1x_1+...+n_tx_t. In this paper we prove that if P is a (poly)matroid base polytope or if P is defined by a TU matrix, then P and projections of P satisfy the integer Caratheodory property.