Abstract

Let P be a simplicial convex d-polytope with fi = fi(P) faces of dimension i. The vector f(P) = (f., fi,..., fdel) is called the f-vector of P. In 1971 McMullen [6; 7, p. 1791 conjectured that a certain condition on a vector f = (f., fi,..., fd...J of integers was necessary and sufficient for f to be the f-vector of some simplicial convex d-polytope. Billera and Lee [l] proved the sufficiency of McMullen’s condition. In this paper we prove necessity. Thus McMullen’s conjecture is completely verified. First we describe McMullen’s condition. Given a simplicial convex d-define polytope P with f(P) = (f., fi,..., f&, where we set fel = 1. The vector h(P) = (h, , h,,..., hd) is called the h-vector of P 181. The Dehn-Sommerville equations, which hold for any simplicial convex polytope, are equivalent to the statement that hi = hdpi, 0 < i,< d [7, Sect. 5.11. If k and i are positive integers, then k can be written uniquely in the form h = (7) + (;“-;) +.. ’ + (q), where ni> ni_r>...> nj> j> 1. Following [6, 8, 91, define h’i ’ = (y+,‘) + (n6-1; 1) +... + (7;;). Also define Oci> = 0. Let us say that a vector (k, , k,,..., Kd) of integers is an M-vector (after F. S. Macaulay) if k, = 1 and 0 < k,.+r < kii ’ for 1 < i, ( d- 1. McMullen’s conjecture may now be stated as follows: A sequence (h, , h,,..., hd) of integers is the h-vector of a simplicial convex d-polytope if and only if h, = 1,

Citations

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