MATHEMATIKA A JOURNAL OF PURE AND APPLIED MATHEMATICS VOL. 17. PART 2. DECEMBER, 1970. No. 34. THE MAXIMUM NUMBERS OF FACES OF A CONVEX POLYTOPE P. McMULLEN 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 c/-polytope with v vertices is achieved by a cyclic polytope CO', d). 1. Introduction. Let P be a c/-polytope, and for 0 < j < d — 1, let/}(P) denote the number of its y-faces. (In matters of terminology, we shall follow Griinbaum [1967] throughout.) Of some practical, as well as considerable theoretical importance is the problem of determining Hj(v, d) = max {fj(P) | P a J-polytope, with/ 0 (P) = v}. The convex hull of v distinct points on the moment curve M, = {(T, T 2 , ..., r d ) e E d | - TO < x < ca}, is called a cyclic polytope C(v, d). The combinatorial type of C(v, d) is independent of the particular choice of the v vertices on M d (see Griinbaum [1967, §4.7]); we write fj(v, d) = fj(C(v, d)). T. S. Motzkin [1957] formulated what has come to be known as THE UPPER-BOUND CONJECTURE. For all I < j < d < v, Hj(v,d)=fj(v,d). In fact, Motzkin's formulation was categorical; however, no proof was subsequently published, and so it seems more reasonable to call the statement a conjecture. Since 1957, considerable efforts have been devoted to proving various cases of the Upper-bound Conjecture. Of particular importance were the contributions of Fieldhouse [1961], Klee [1964b] and Gale [1964]; it should also be noted that the cases d < 4 were already known to Bruckner [1893]. For detailed discussions of the history of these investigations, the reader should consult Griinbaum [1967, §10.1, 1970]. [MATHEMATIKA 17 (1970), 179-184] available at https://www.cambridge.org/core/terms. https://doi.org/10.1112/S0025579300002850 Downloaded from https://www.cambridge.org/core. IP address: 54.162.69.248, on 18 Jun 2020 at 06:14:20, subject to the Cambridge Core terms of use,