Lower Bounds on Face Numbers of Polytopes with $m$ Facets

Abstract

Let $P$ be a convex $d$-polytope and $0 \leq k \leq d-1$. In 2023, this author proved the following inequalities, resolving a question of Bárány:
\[
\frac{f_k(P)}{f_0(P)} \geq
\frac{1}{2}\biggl[\binom{\lceil \frac{d}{2} \rceil}{k} +
\binom{\lfloor \frac{d}{2} \rfloor}{k}\biggr],
\qquad
\frac{f_k(P)}{f_{d-1}(P)} \geq \frac{1}{2}\biggl[\binom{\lceil \frac{d}{2} \rceil}{d-k-1} + \binom{\lfloor \frac{d}{2} \rfloor}{d-k-1}\biggr].
\]

We show that for any fixed $d$ and $k$, these are the tightest possible linear bounds on $f_k(P)$ in terms of $f_0(P)$ or $f_{d-1}(P)$. We then give a stronger bound on $f_k(P)$ in terms of the Grassmann angle sum $\gamma_k^2(P)$. Finally, we prove an identity relating the face numbers of a polytope with the behavior of its facets under a fixed orthogonal projection of codimension two.