A Lower Bound Theorem for Strongly Regular CW Spheres with up to 2d+1 Vertices.

In 1967, Grünbaum conjectured that any d-dimensional polytope with d + s ⩽ 2 d vertices has at least ϕ k (d + s , d) = d + 1 k + 1 + d k + 1 - d + 1 - s k + 1 k-faces. This conjecture along with the characterization of equality cases was recently proved by the author (A proof of Grünbaum's lower bound conjecture for general polytopes. Israel J. Math. 245(2), 991–1000 (2021)). In this paper, several extensions of this result are established. Specifically, it is proved that lattices with the diamond property (e.g., abstract polytopes) and d + s ⩽ 2 d atoms have at least ϕ 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 2 d + 1 vertices are obtained. These bounds are given by the face numbers of certain polytopes with 2 d + 1 vertices. [ABSTRACT FROM AUTHOR]