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

TL;DR

This paper refines a lower bound theorem for $d$-polytopes with at most $2d$ vertices, identifying minimal face counts and characterizing the polytopes that achieve these bounds.

Contribution

It extends Xue's theorem to polytopes with more facets, providing precise counts and characterizations of minimising polytopes for various parameters.

Findings

Unique minimisers for many cases when s=2

Lower bounds on the number of k-faces for various s and k

Characterization of the polytopes that attain these bounds

Abstract

In 1967, Gr\"unbaum conjectured that the function $$ \phi_k(d+s,d):=\binom{d+1}{k+1}+\binom{d}{k+1}-\binom{d+1-s}{k+1},\; \text{for $2\le s\le d$} $$ 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 $k\in[1\ldots d-2]$ 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 $\phi_k(d+s,d)+\binom{d-1}{k}-\binom{d+1-s}{k}$, 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.