A lower bound theorem for $d$-polytopes with $2d+1$ vertices

TL;DR

This paper extends the understanding of lower bounds on the number of faces of $d$-polytopes with $n$ vertices, specifically establishing results for the case where $n=2d+1$, and characterizes polytopes with minimal vertices and edges.

Contribution

It provides the first lower bound theorem for $d$-polytopes with $2d+1$ vertices, revealing different bounds and minimizers compared to the $n extless=2d$ case, and characterizes polytopes with $d+3$ vertices.

Findings

Established lower bounds for $d$-polytopes with $2d+1$ vertices.

Characterized all $d$-polytopes with $d+3$ vertices and minimal edges.

Identified differences in the nature of bounds and minimizers from previous cases.

Abstract

The problem of calculating exact lower bounds for the number of $k$-faces of $d$-polytopes with $n$ vertices, for each value of $k$, and characterising the minimisers, has recently been solved for $n\le2d$. We establish the corresponding result for $n=2d+1$; the nature of the lower bounds and the minimising polytopes are quite different in this case. As a byproduct, we also characterise all $d$-polytopes with $d+3$ vertices, and only one or two edges more than the minimum.