Improving bounds on the diameter of a polyhedron in high dimensions

TL;DR

This paper improves existing theoretical bounds on the maximum diameter of high-dimensional polyhedra, refining previous results to tighter bounds that depend on the number of facets and the dimension.

Contribution

The authors significantly tighten the upper bounds on polyhedron diameters, advancing the theoretical understanding of polyhedral geometry in high dimensions.

Findings

Improved bound to $(n-d)^{-1+ ext{log}_2 d}$ for $n \\ge d \\ge 7$

Enhanced bound to $(n-d)^{-2+ ext{log}_2 d}$ for $n \\ge d \\ge 37$

Refined bound to $(n-d)^{-3+ ext{log}_2 d+O(1/d)}$ for $n \\ge d \\ge 1

Abstract

In 1992, Kalai and Kleitman proved that the diameter of a $d$-dimensional polyhedron with $n$ facets is at most $n^{2+\log_2 d}$. In 2014, Todd improved the Kalai-Kleitman bound to $(n-d)^{\log_2 d}$. We improve the Todd bound to $(n-d)^{-1+\log_2 d}$ for $n \ge d \ge 7$, $(n-d)^{-2+\log_2 d}$ for $n \ge d \ge 37$, and $(n-d)^{-3+\log_2 d+O\left(1/d\right)}$ for $n \ge d \ge 1$.