The maximum numbers of faces of a convex polytope

Abstract

In this paper we give a proof of the long‐standing Upper‐bound Conjecture for convex polytopes, which states that, for 1 ≤ j < d < v, the maximum possible number of j‐faces of a d‐polytope with v vertices is achieved by a cyclic polytope C(v, d).

Newton polyhedra (algebra and geometry)

“…What is the maximum possible number of k-dimensional faces in a simple l-dimensional polyhedron which has p faces of the highest dimension? The answer to this question is given by the famous Upper-Bound Theorem, proved in 1970 by McMullen [21]. Every simple l-dimensional polyhedron with p faces of the highest dimension can be considered as a generic i-dimensional section of a (p − 1)-dimensional simplex.…”

Approximating Joint Probability Distributions Given Partial Information

“…For example, consider a simple joint probability distribution comprised of eight binary random variables, whose marginal distributions are known. The polytope encoding these constraints could have up to 10 13 vertices (McMullen 1970). Although this is an upper bound, the number of vertices likely to present in real-world problems is still enormous (Schmidt and Mattheiss 1977).…”

Empty Simplices of Polytopes and Graded Betti Numbers

“…According to McMullen's Upper Bound Theorem ( [13]), the cyclic polytope C(f 0 , d) has the maximal f -vector among all simplicial d-polytopes with f 0 vertices. Theorem 4.15 shows that it also has the maximal total number of empty simplices among these polytopes.…”