Tail diameter upper bounds for polytopes and polyhedra

TL;DR

This paper establishes new tail bounds, including quasipolynomial and polynomial upper bounds, on the diameters of polytopes, polyhedra, and related complexes, advancing understanding of their geometric properties.

Contribution

It introduces tail-quasipolynomial and tail-polynomial upper bounds on diameters, extending prior bounds to broader classes of polytopes and complexes.

Findings

Tail-quasipolynomial bounds for polytopes and complexes

Tail-polynomial bounds for polyhedra

Extension of diameter bounds beyond previous limitations

Abstract

In 1992, Kalai and Kleitman proved a quasipolynomial upper bound on the diameters of convex polyhedra. Todd and Sukegawa-Kitahara proved tail-quasipolynomial bounds on the diameters of polyhedra. These tail bounds apply when the number of facets is greater than a certain function of the dimension. We prove tail-quasipolynomial bounds on the diameters of polytopes and normal simplicial complexes. We also prove tail-polynomial upper bounds on the diameters of polyhedra.