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

Let $P$ be a convex $d$-polytope and $0\le k\le d-1$. In 2023, this author proved the following inequalities, resolving a question of Bárány: $$\frac{f_k(P)}{f_0(P)}\ge \frac{1}{2}[\left(\genfrac{}{}{0px}{}{\lceil \frac{d}{2}\rceil }{k}\right)+\left(\genfrac{}{}{0px}{}{\lfloor \frac{d}{2}\rfloor }{k}\right)],\frac{f_k(P)}{f_{d-1}(P)}\ge \frac{1}{2}[\left(\genfrac{}{}{0px}{}{\lceil \frac{d}{2}\rceil }{d-k-1}\right)+\left(\genfrac{}{}{0px}{}{\lfloor \frac{d}{2}\rfloor }{d-k-1}\right)]\mathrm{.}$$ 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.