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 $$ f_r(S(n_1,d))\le f_r(P) \le f_r(C(n_2,d)) $$ for some integers $n_1, n_2$ and $r$, then $$ f_s(S(n_1,d))\le f_s(P) \le f_s(C(n_2,d)) $$ 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. 
Anders Bj\"orner