 
Summary: matroid polytopes and their volumes.
federico ardila
carolina benedetti
jeffrey doker
abstract.
We express the matroid polytope PM of a matroid M as a signed Minkowski
sum of simplices, and obtain a formula for the volume of PM . This gives a
combinatorial expression for the degree of an arbitrary torus orbit closure
in the Grassmannian Grk,n. We then derive analogous results for the inde
pendent set polytope and the associated flag matroid polytope of M. Our
proofs are based on a natural extension of Postnikov's theory of generalized
permutohedra.
1 introduction.
The theory of matroids can be approached from many different points of view; a
matroid can be defined as a simplicial complex of independent sets, a lattice of
flats, a closure relation, etc. A relatively new point of view is the study of matroid
polytopes, which in some sense are the natural combinatorial incarnations of
matroids in algebraic geometry and optimization. Our paper is a contribution in
this direction.
We begin with the observation that matroid polytopes are members of the
