Let \(\mathscr{X}\) be the boundary complex of a (d+1)-polytope, and let \(\rho (d + 1,k) = \frac{1}{2}\left[ {\left( {\begin{array}{*{20}{c}} {\left\lceil {(d + 1)/2} \right\rceil } \\ {d - k} \end{array}} \right)} \right] + \left[ {\left( {\begin{array}{*{20}{c}} {\left\lfloor {(d + 1)/2} \right\rfloor } \\ {d - k} \end{array}} \right)} \right]\) . Recently, the author, answering Bárány’s question from 1998, proved that for all \(\lfloor {{{d - 1} \over 2}}\rfloor \le k \le d\) \(f_{k}({\mathscr{X}})\ge \rho(d+1,k)f_{d}(\mathscr{X}).\)
We prove a generalization: if \({\mathscr{X}}\) is a shellable, strongly regular CW sphere or CW ball of dimension d, then for all \(\lfloor {{{d - 1} \over 2}} \rfloor \le k \le d\) \(f_{k}({\mathscr{X}})\ge \rho(d+1,k)f_{d}({\mathscr{X}})+{1\over 2}f_{k}(\partial\mathscr{X}),\) 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 \(f_{k}({\mathscr{S}})\ge\text{min}\{f_{0}({\mathscr{S}}),f_{d}({\mathscr{S}})\}\) for all 0 ≤ k ≤ d.
