A Simple Proof of the Upper Bound Theorem

Let c i ( n, d ) be the number of i -dimensional faces of a cyclic d -polytope on n vertices. We present a simple new proof of the upper bound theorem for convex polytopes, which asserts that the number of i -dimensional faces of any d -polytope on n vertices is at most c i ( n, d ). Our proof applies for arbitrary shellable triangulations of ( d −1) spheres. Our method provides also a simple proof of the upper bound theorem for d -representable complexes.