More bounds on the diameters of convex polytopes

TL;DR

This paper investigates the maximum edge diameter of convex polytopes, providing new bounds and exact values that challenge existing conjectures, especially showing the Hirsch bound is not always tight in dimension 4.

Contribution

The authors establish the exact value of (4,12)=7 and provide evidence that (5,12)=(6,13)=7, advancing understanding of polytope diameters and the Hirsch conjecture.

Findings

(4,12)=7

(5,12)=7

(6,13)=7

Abstract

Finding a good bound on the maximal edge diameter $\Delta(d,n)$ of a polytope in terms of its dimension $d$ and the number of its facets $n$ is one of the basic open questions in polytope theory \cite{BG}. Although some bounds are known, the behaviour of the function $\Delta(d,n)$ is largely unknown. The Hirsch conjecture, formulated in 1957 and reported in \cite{GD}, states that $\Delta(d,n)$ is linear in $n$ and $d$: $\Delta(d,n) \leq n-d$. The conjecture is known to hold in small dimensions, i.e., for $d \leq 3$ \cite{VK}, along with other specific pairs of $d$ and $n$ (Table \ref{before}). However, the asymptotic behaviour of $\Delta(d,n)$ is not well understood: the best upper bound -- due to Kalai and Kleitman -- is quasi-polynomial \cite{GKDK}. In this article we will show that $\Delta(4,12)=7$ and present strong evidence for $\Delta(5,12)=\Delta(6,13)=7$. The first of these new values is of particular interest since it indicates that the Hirsch bound is not sharp in dimension 4.