  Let $P$ be a polytope. The hitting number of $P$ is the smallest size of a
hitting set of the facets of $P$, i.e., a subset of vertices of $P$ such that
every facet of $P$ has a vertex in the subset. An extended formulation of $P$
is the description of a polyhedron that linearly projects to $P$. We show that,
if $P$ is the base polytope of any matroid, then $P$ admits an extended
formulation whose size depends linearly on the hitting number of $P$. Our
extended formulations generalize those of the spanning tree polytope given by
Martin and Wong. Our proof is simple and short, and it goes through the deep
connection between extended formulations and communication protocols.
