Let ci(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 ci(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.
All Science Journal Classification (ASJC) codes
- Discrete Mathematics and Combinatorics