A COMPARISON THEOREM FOR f-VECTORS OF SIMPLICIAL POLYTOPES

Let S(n, d) and C(n, d) denote, respectively, the stacked and the cyclic d-dimensional polytopes on n vertices. Furthermore, fi(P ) denotes the number of i-dimensional faces of a polytope P . Theorem 1. Let P be a d-dimensional simplicial polytope. Suppose that fr(S(n1, d)) ≤ fr(P ) ≤ fr(C(n2, d)) for some integers n1, n2 and r ≤ d − 2. Then, fs(S(n1, d)) ≤ fs(P ) ≤ fs(C(n2, d)) for all s such that r < s < d. Some special cases were previously known. For r = 0 these inequalities are the well-known lower and upper bound theorems for simplicial polytopes, see e.g. [7, Ch. 8]. The s = d−1 case of the upper bound part is contained in the “generalized upper bound theorem” of Kalai [3, Theorem 2]. The result is implied by a more general “comparison theorem” for f -vectors, formulated in Section 1. Among its other consequences is a similar lower bound theorem for centrally-symmetric simplicial polytopes.