A refined lower bound theorem for $d$-polytopes with at most $2d$ vertices

Abstract

In 1967, Grünbaum conjectured that the function $$\ddot{I}{•}_k(d+s,d):=\left(\genfrac{}{}{0ex}{}{d+1}{k+1}\right)+\left(\genfrac{}{}{0ex}{}{d}{k+1}\right)-\left(\genfrac{}{}{0ex}{}{d+1-s}{k+1}\right),\text{for 2≤s≤d}$$ provides the minimum number of $k$-faces for a $d$-dimensional polytope (abbreviated as a $d$-polytope) with $d+s$ vertices. In 2021, Xue proved this conjecture for each $k\in [1\dots d-2]$ and characterised the unique minimisers, each having $d+2$ facets. In this paper, we refine Xue's theorem by considering $d$-polytopes with $d+s$ vertices ($2\le s\le d$) and at least $d+3$ facets. If $s=2$, then there is precisely one minimiser for many values of $k$. For other values of $s$, the number of $k$-faces is at least $\ddot{I}{•}_k(d+s,d)+\left(\genfrac{}{}{0ex}{}{d-1}{k}\right)-\left(\genfrac{}{}{0ex}{}{d+1-s}{k}\right)$, which is met by precisely two polytopes in many cases, and up to five polytopes for certain values of $s$ and $k$. We also characterise the minimising polytopes.

31 pages, 2 figures