A Proof of Grünbaum’s Lower Bound Conjecture for general polytopes

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

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In 1967, Grünmbaum conjectured that any d-dimensional polytope with d + s ≤ 2d vertices has at leastk-faces. This conjecture along with the characterization of equality cases was recently proved by the author [18]. In this paper, several extensions of this result are established.

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“…k-faces. This conjecture along with the characterization of equality cases was recently proved by the author [18]. In this paper, several extensions of this result are established.…”

“…Theorem 3.2 of [18] established Grünbaum's conjecture for all polytopes. The goal of this section is to extend this result to a much larger class of objects, namely, to all diamond lattices.…”

“…The conjecture remained completely open for s ≥ 5 until Pineda-Villavicencio, Ugon and Yost [13] proved this conjecture for the number of edges, i.e., they verified the k = 1 case. In 2020, we proved (see [18]) the conjecture in full generality and characterized equality cases.…”

A Positive Answer to Bárány's Question on Face Numbers of Polytopes

“…It is well known that for a fixed dimension d, the d-simplex ∆ d simultaneously minimizes all face numbers. The most general lower bound result to date is a recent theorem of Xue [14], originally conjectured by Grünbaum in 1967 [7]…”