The polytope of all matroids

Abstract: It is possible to write the indicator function of any matroid polytope as an integer combination of indicator functions of Schubert matroid polytopes. In this way, every matroid on $n$ elements of rank $r$ can be thought of as a lattice point in the space having a coordinate for each Schubert matroid on $n$ elements of rank $r$. We study the convex hull of all these lattice points, with particular focus on the vertices, which come from the matroids we call extremal matroids. We show that several famous classes of matroids arise as faces of the polytopes, and in many cases we determine the dimension of this face explicitly. As an application, we show that there exist valuative invariants that attain non-negative values at all representable matroids, but fail to be non-negative in general.