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

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.

35 pages. With a new co-author, Jie Wang, we have strengthened our results to include an equality condition: if a $d$-polytope $P$ with $2d+2$ vertices and at least $d+3$ facets attains the lower bound for some $1\le k\le d-2$, then $P$ has exactly $d+3$ facets