Combinatorics Seminar:A Proof of Grünbaum's Lower Bound Conjecture for Polytopes, Lattices, and Strongly Regular Pseudomanifolds

Abstract: If we fix the dimension and the number of vertices of a polytope, what is the smallest number of faces of each dimension? In 1967, Grünbaum made a conjecture on this lower bound problem for $d$-dimensional polytopes with at most $2d$ vertices. In the talk, we will discuss the proof of this conjecture and the equality cases. We will then extend our results to lattices with diamond property (the inequality part) and to strongly regular normal pseudomanifolds (the equality part). We will also talk about recent results on $d$-dimensional polytopes with $2d+1$ or $2d+2$ vertices.

Host: Jodi McWhirter