Counting the interior points of a point configuration

We prove a formula conjectured by Ahrens, Gordon, and McMahon for the numbe r of interior points for a point configuration in R-d. Our method is to sho w that the formula can be interpreted as a sum of Euler characteristics of certain complexes associated with the point configuration, and then compute the homology of these complexes. This method extends to other examples of convex geometries. We sketch these applications, replicating an earlier res ult of Gordon, and proving a new result related to ordered sets.