  We define the root polytope of a regular oriented matroid, and show that the
greedoid polynomial of an Eulerian branching greedoid rooted at vertex $v_0$ is
equivalent to the $h^*$-polynomial of the root polytope of the dual of the
graphic matroid.
  As the definition of the root polytope is independent of the vertex $v_0$,
this gives a geometric proof for the root-independence of the greedoid
polynomial for Eulerian branching greedoids, a fact which was first proved by
Swee Hong Chan, K\'evin Perrot and Trung Van Pham using sandpile models. We
also obtain that the greedoid polynomial does not change if we reverse every
edge of an Eulerian digraph.
