Matroid Polytopes and their Volumes
We express the matroid polytope P (M) of a matroid M as a signed Minkowski sum of simplices, and obtain a formula for the volume of P (M) . This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian Gr (k,n) . We then derive analogous results for the independent set polytope and the underlying flag matroid polytope of M. Our proofs are based on a natural extension of Postnikov's theory of generalized permutohedra.
matroid matroid polytope minkowski sum generalized permutahedron