  In 1980, Las Vergnas defined a notion of discrete convexity for oriented
matroids, which Edelman subsequently related to the theory of anti-exchange
closure functions and convex geometries. In this paper, we use generalized
matroid activity to construct a convex geometry associated with an ordered,
unoriented matroid. The construction in particular yields a new type of
representability for an ordered matroid defined by the affine representability
of its corresponding convex geometry.
  The lattice of convex sets of this convex geometry induces an ordering on the
matroid independent sets which extends the external active order on matroid
bases. We show that this generalized external order forms a supersolvable
meet-distributive lattice refining the geometric lattice of flats, and we
uniquely characterize the lattices isomorphic to the external order of a
matroid. Finally, we introduce a new trivariate generating function
generalizing the matroid Tutte polynomial.
