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

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 B\'ar\'any: $$\frac{f_k(P)}{f_0(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose k} + {\lfloor \frac{d}{2} \rfloor \choose k}\biggr], \qquad \frac{f_k(P)}{f_{d-1}(P)} \geq \frac{1}{2}\biggl[{\lceil \frac{d}{2} \rceil \choose d-k-1} + {\lfloor \frac{d}{2} \rfloor \choose d-k-1}\biggr].$$ 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.