Face Numbers of Shellable CW Balls and Spheres

Let $\mathcal{X}$ be the boundary complex of a $(d+1)$-polytope, and let $\rho (d+1,k)=\frac{1}{2}[\left(\genfrac{}{}{0px}{}{\lceil (d+1)\mathrm{/}2\rceil }{d-k}\right)+\left(\genfrac{}{}{0px}{}{\lfloor (d+1)\mathrm{/}2\rfloor }{d-k}\right)]$. Recently, the author, answering Bárány’s question from 1998, proved that for all $\lfloor \frac{d-1}{2}\rfloor \le k\le d$, $$f_k(\mathcal{X})\ge \rho (d+1,k)f_d(\mathcal{X})\mathrm{.}$$ We prove a generalization: if $\mathcal{X}$ is a shellable, strongly regular CW sphere or CW ball of dimension $d$, then for all $\lfloor \frac{d-1}{2}\rfloor \le k\le d$, $$f_k(\mathcal{X})\ge \rho (d+1,k)f_d(\mathcal{X})+\frac{1}{2}{f}_k(\mathrm{\partial }\mathcal{X}),$$ with equality precisely when $k=d$ or when $k=d-1$ and $\mathcal{X}$ is simplicial. We further prove that if $\mathcal{S}$ is a strongly regular CW sphere of dimension $d$, and the face poset of $\mathcal{S}$ is both CL-shellable and dual CL-shellable, then $f_k(\mathcal{S})\ge \min \{f_0(\mathcal{S}),f_d(\mathcal{S})\}$ for all $0\le k\le d$.