Counting polytopes via the Radon complex

A convex polytope is the convex hull of a finite set of points. We introduce the Radon complex of a polytope—a subcomplex of an appropriate hypercube which encodes all Radon partitions of the polytopeʹs vertex set. By proving that such a complex, when the vertices of the polytope are in general position, is homeomorphic to a sphere, we find an explicit formula to count the number of d-dimensional polytope types with d+3 vertices in general position.