The Upper Bound Theorem for Polytopes: an Easy Proof of Its Asymptotic Version

Since at least half of the d edges incident to a vertex u of a simple d-polytope P either all point “up” or all point “down,” v must be the unique “bottom” or “top” vertex of a face of P of dimension at least d/2. Thus the number of P’s vertices is at most twice the number of such high-dimensional faces, which is at most Ed,2 $ iG &Ynu) = O(nld/‘l), if P has n facets. This, in a nutshell, provides a proof of the asymptotic version of the famous upper bound theorem: A convex d-polytope with n facets (or, dually, with n vertices) has O(r~l~/~l) faces, when d is constant. 1. Remarks and discussion For a d-polytope P let f,(P) denote the number of i-faces of P. How large can f,(P) get, provided fd_ ,(P> = 12, i.e. P has 12 facets? In 1970 McMullen [5] proved the exact bound of n-l-max(j, d-j) min{j, d-j) ’ after Klee [4], [3, sec. 10.11 had proven it for sufficiently large values of n. Their proofs relied on the so-called Dehn-Sommerville relations and in McMullen’s case on the existence of shellings. * Supported by NSF Presidential Young Investigator award CCR-9058440. Email: [email protected] 09257721/95/$09.50