The upper bound theorem for polytopes: an easy proof of its asymptotic version

✍ Scribed by Raimund Seidel

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 128 KB
• Volume: 5
• Category: Article
• ISSN: 0925-7721
• DOI: 10.1016/0925-7721(95)00013-y

✦ Synopsis

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.