The p-negative type behaviour of finite metric spaces

Abstract

In this thesis we examine the p-negative type behaviour of finite metric spaces. Previous work done in the settings of p-negative type, generalised roundness p and quasihypermetricity, by various authors, is integrated and unified. These various approaches are used to deduce a practical formula for the supremal p-negative type of a given finite metric space based on its matrix of pair-wise distances. This allows us to prove results about the $p$-negative type behaviour of some path metric graphs, and perform experiments from which conjectures can be made. It also allows us to find the critical vectors demonstrating the supremal p-negative type value, and we prove several symmetry results about them.

A class of metric spaces called p-additive combination spaces is introduced and shown to have interesting p-negative type properties. In particular, the formula for the 1-negative type gap for finite metric trees of Doust-Weston is shown to greatly generalise to p-additive combination spaces and a completely new proof of the Doust-Weston result is found. The p-negative type behaviour of finite ultrametric spaces is then examined, with a neat combinatorial formula found for the asymptotic behaviour of the p-negative type gap. We finish by presenting some conjectures naturally arising from our work, and potential avenues of progress.