Let denote the number of -faces of a -polytope , for .
The exact upper bound for in terms of and . is known, thanks to McMullen's upper bound theorem.
The convex hull of distinct points on the moment curve in is known as a cyclic polytope. It is known that its combinatorial structure (i.e. its face lattice, see Section 2.3) is uniquely determined by and . Thus we often write to denote any such cyclic -polytope with vertices.
McMullen's Upper Bound Theorem shows that the maximum of is attained by the cyclic polytopes.
The number of -faces of a cyclic polytope can be explicitely given and thus one can evaluate the order of the upper bound in terms of and .
The upper bound theorem can be written in dual form which gives, for example, the maximum number of vertices in a -polytope with facets.
The original proof of the Upper Bound Theorem is in [McM70,MS71]. There are different variations, see [Kal97,Mul94,Zie94].