Enumeration in convex geometries and associated polytopal subdivisions   of spheres
  We construct CW spheres from the lattices that arise as the closed sets of a convex closure, the meet-distributive lattices. These spheres are nearly polytopal, in the sense that their barycentric subdivisions are simplicial polytopes. The complete information on the numbers of faces and chains of faces in these spheres can be obtained from the defining lattices in a manner analogous to the relation between arrangements of hyperplanes and their underlying geometric intersection lattices. 
Louis J. Billera, Samuel K. Hsiao, J. Scott Provan