Coxeter matroid polytopes

If ∆ is a polytope in real affine space, each edge of ∆ determines a reflection in the perpendicular bisector of the edge. The exchange group W (∆) is the group generated by these reflections, and ∆ is a (Coxeter) matroid polytope if this group is finite. This simple concept of matroid polytope turns out to be an equivalent way to define Coxeter matroids. The GelfandSerganova Theorem and the structure of the exchange group both give us information about the matroid polytope. We then specialise this information to the case of ordinary matroids to learn more about the classical matroid polytope already familiar to matroid theorists. Definitions and notation mostly follow [H] for reflection groups and Coxeter groups, [O, Wel, Wh] for matroids, [BG, BR, BGW1] for Coxeter matroids.