A lower bound theorem for $d$-polytopes with $2d+2$ vertices

TL;DR

This paper establishes new lower bounds on the number of $k$-faces in $d$-dimensional polytopes with $2d+2$ vertices, revealing how the number of facets influences these bounds and identifying minimisers for small dimensions.

Contribution

It extends lower bound theorems for $k$-faces to polytopes with $2d+2$ vertices, detailing the role of facet count in tight bounds and characterizing minimisers.

Findings

Two distinct lower bounds depending on the number of facets.

Exact minimisers identified for dimensions up to 5.

Lower bounds are tight under specific facet conditions.

Abstract

We establish a lower bound theorem for the number of $k$-faces ($1\le k\le d-2$) in a $d$-dimensional polytope $P$ (abbreviated as a $d$-polytope) with $2d+2$ vertices, extending the previously known case for $k=1$. We identify all minimisers for $d\le 5$. Two distinct lower bounds emerge, depending on the number of facets of $P$. When $P$ has precisely $d+2$ facets, the lower bound is tight when $d$ is odd. If $P$ has at least $d+3$ facets, the lower bound is always tight, and equality holds for some $1\le k\le d-2$ only when $P$ has precisely $d+3$ facets. Moreover, for $1\le k\le \ceil{d/3}-2$, the minimisers among $d$-polytopes with $2d+2$ vertices have precisely $d+3$ facets, while for $\floor{0.4d}\le k\le d-1$, the lower bound arises from $d$-polytopes with $d+2$ facets.