Face Numbers of Shellable CW Balls and Spheres

Let $\mathscr{X}$ be the boundary complex of a $(d+1)$-polytope, and let $\rho(d+1,k) = \frac{1}{2}[{\lceil (d+1)/2 \rceil \choose d-k} + {\lfloor (d+1)/2 \rfloor \choose d-k}]$. Recently, the author, answering B\'ar\'any's question from 1998, proved that for all $\lfloor \frac{d-1}{2} \rfloor \leq k \leq d$, $$f_k(\mathscr{X}) \geq \rho(d+1,k)f_d(\mathscr{X}).$$ We prove a generalization: if $\mathscr{X}$ is a shellable, strongly regular CW sphere or CW ball of dimension $d$, then for all $\lfloor \frac{d-1}{2} \rfloor \leq k \leq d$, $$f_k(\mathscr{X}) \geq \rho(d+1,k)f_d(\mathscr{X}) + \frac{1}{2}f_k(\partial \mathscr{X}),$$ with equality precisely when $k=d$ or when $k=d-1$ and $\mathscr{X}$ is simplicial. We further prove that if $\mathscr{S}$ is a strongly regular CW sphere of dimension $d$, and the face poset of $\mathscr{S}$ is both CL-shellable and dual CL-shellable, then $f_k(\mathscr{S}) \geq \min\{f_0(\mathscr{S}),f_d(\mathscr{S})\}$ for all $0 \leq k \leq d$.