A Lower Bound Theorem for strongly regular CW spheres with up to $2d+1$ vertices

In 1967, Gr\"unmbaum conjectured that any $d$-dimensional polytope with $d+s\leq 2d$ vertices has at least $$\phi_k(d+s,d) = {d+1 \choose k+1 }+{d \choose k+1 }-{d+1-s \choose k+1 }$$ $k$-faces. This conjecture along with the characterization of equality cases was recently proved by the author. In this paper, several extensions of this result are established. Specifically, it is proved that lattices with the diamond property (for example, abstract polytopes) and $d+s\leq 2d$ atoms have at least $\phi_k(d+s,d)$ elements of rank $k+1$. Furthermore, in the case of face lattices of strongly regular CW complexes representing normal pseudomanifolds with up to $2d$ vertices, a characterization of equality cases is given. Finally, sharp lower bounds on the number of $k$-faces of strongly regular CW complexes representing normal pseudomanifolds with $2d+1$ vertices are obtained. These bounds are given by the face numbers of certain polytopes with $2d+1$ vertices.