Upper bound theorem

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.

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.