A comparison theorem for f-vectors of simplicial polytopes

Let f(i)(P) denote the number of i-dimensional faces of a convex polytope P. Furthermore, let S(n, d) and C(n, d) denote, respectively, the stacked and the cyclic d-dimensional polytopes on n vertices. Our main result is that for every simplicial d-polytope P, if f(r) (S (n(1), d)) <= f(r) (P) <= f(r) (C (n(2), d)) for some integers n(1), n(2) and r, then f(s) (S (n(1), d)) <= f(s) (P) <= f(s) (C (n(2), d)) for all s such that r < s. For r = 0 these inequalities are the well-known lower and upper bound theorems for simplicial polytopes. The result is implied by a certain "comparison theorem" for f-vectors, formulated in Section 4. Among its other consequences is a similar lower bound theorem for centrally-symmetric simplicial polytopes.