McMullen's Conditions and Some Lower Bounds for General Convex Polytopes

Convex polytopes are the d-dimensional analogues of 2-dimensional convex polygones and 3-dimensional convex polyhedra. A polytope is a bounded convex set in R d that is the intersection of a finite number of closed halfspaces. The faces of a polytope are its intersections with supporting hyperplanes. The/-dimensional faces are called the i-faces and f i ( P ) denotes the number of i-faces of a polytope P; the d-tuple ( fo(P) , f l ( P ) , . . . , f d l ( P ) ) i s called the f-vector of P. In particular, 0faces, 1-faces and ( d 1)-faces are respectively called vertices, edges and facets of a d-dimensional polytope. One of the most important question in the combinatorial theory of convex polytopes is the determination of the largest and the smallest number of/-faces of a d-dimensional polytope with a given number of k-faces. Moreover, it is also interesting to find out which class of polytopes attains these bounds. General references to the topics discussed in our paper are [5], [6], [9]. In this section we first recall McMullen's upper bound theorem and Barnette's lower bound theorem for simplicial polytopes. Then we present our lower bounds for general convex polytopes. The upper bound theorem was conjectured by Motzkin [10] in 1957 and proved by McMullen [7] in 1970. In order to state this theorem, we define for i >_ 0: