Abstract:

After an introduction to and review of the upper and lower bound theorems for simplicial polytopes (due to McMullen and Barnette, respectively, circa 1970), I will go on to discuss the following theorem:

Let $S(n,d)$ and $C(n,d)$ denote, respectively, the stacked and the cyclic $d$-dimensional polytopes on $n$ vertices. Furthermore, $f_i(P)$ denotes the number of $i$-dimensional faces of a polytope $P$.

\begin{thm}
Let $P$ be a $d$-dimensional simplicial polytope.\\
Suppose that
$$f_r(S(n_1,d))\le f_r(P) \le f_r(C(n_2,d))$$
for some integers $n_1, n_2$ and $r\le d-2$. Then,
$$f_s(S(n_1,d))\le f_s(P) \le f_s(C(n_2,d))$$
for all $s$ such that $r<s<d$.
\end{thm}

Some special cases were previously known. For $r=0$ these inequalities are the well-known lower and upper bound theorems. The $s=d-1$ case of the upper bound part is the "generalized upper bound theorem" of Kalai.

The result is implied by a more general "comparison theorem" for $f$-vectors. Among its consequences is a similar lower bound theorem for centrally-symmetric simplicial polytopes.