A lower bound theorem for $d$-polytopes with at most $3d-1$ vertices

TL;DR

This paper establishes a new lower bound theorem for the number of certain faces in $d$-dimensional polytopes with up to $3d-1$ vertices, extending previous bounds for fewer vertices.

Contribution

It introduces a lower bound theorem for $d$-polytopes with up to $3d-1$ vertices, filling a gap in the understanding of face counts for these polytopes.

Findings

Lower bounds are tight for polytopes with $d+2$ facets and $2d+ ext{(some } ext{ell})$ vertices.

The bounds remain tight up to $ ext{ell}=d-1$ for polytopes with at least $d+3$ facets.

Explicit minimisers are exhibited for each number of vertices between $2d+1$ and $3d-1$.

Abstract

We prove a lower bound theorem for the number of $k$-faces ($1\le k\le d-2$) in a $d$-dimensional polytope $P$ (or $d$-polytope) with up to $3d-1$ vertices. Previous lower bound theorems for $d$-polytopes with few vertices concern those with at most $2d$ vertices, $2d+1$ vertices, and $2d+2$ vertices. If $P$ has exactly $d+2$ facets and $2d+\ell$ vertices ($\ell\ge 1$), the lower bound is tight for certain combinations of $d$ and $\ell$. When $P$ has at least $d+3$ facets and $2d+\ell$ vertices ($\ell\ge 1$), the lower bound remains tight up to $\ell=d-1$, and equality for some $1\le k\le d-2$ is attained only when $P$ has precisely $d+3$ facets. We exhibit at least one minimiser for each number of vertices between $2d+1$ and $3d-1$, including two distinct minimisers with $2d+2$ vertices and three with $3d-2$ vertices.