In mathematics, the upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of vertices. It is one of the central results of polyhedral combinatorics.
Contents
Originally known as the upper bound conjecture, this statement was formulated by Theodore Motzkin, proved in 1970 by Peter McMullen, and strengthened from polytopes to subdivisions of a sphere in 1975 by Richard P. Stanley.
Cyclic polytopes
The cyclic polytope Δ(n,d) may be defined as the convex hull of n vertices on the moment curve (t, t2, t3, ...). The precise choice of which n points on this curve are selected is irrelevant for the combinatorial structure of this polytope. The number of i-dimensional faces of Δ(n,d) is given by the formula
and
Statement
The upper bound theorem states that if Δ is a simplicial sphere of dimension d − 1 with n vertices, then
That is, the number of faces of an arbitrary polytope can never be more than the number of faces of a cyclic or neighborly polytope with the same dimension and number of vertices. Asymptotically, this implies that there are at most
History
The upper bound conjecture for simplicial polytopes was proposed by Motzkin in 1957 and proved by McMullen in 1970. A key ingredient in his proof was the following reformulation in terms of h-vectors:
Victor Klee suggested that the same statement should hold for all simplicial spheres and this was indeed established in 1975 by Stanley using the notion of a Stanley–Reisner ring and homological methods. For a nice historical account of this theorem see.