Strict p-negative type of a metric space

TL;DR

This paper refines a technique to better estimate the strict p-negative type of finite metric spaces, showing that the supremal p-negative type cannot be strict and classifying the intervals where strict p-negative type occurs.

Contribution

It improves bounds on the supremal strict p-negative type for finite metric spaces and proves that this supremum cannot be strict, extending previous methods to more general spaces.

Findings

Improved lower bounds for finite metric trees' p-negative type.

The supremal p-negative type of a finite metric space cannot be strict.

Finite metric spaces with p-negative type for p > 0 have strict q-negative type for all q in [0,p).

Abstract

Doust and Weston introduced a new method called "enhanced negative type" for calculating a non trivial lower bound p(T) on the supremal strict p-negative type of any given finite metric tree (T,d). In the context of finite metric trees any such lower bound p(T) > 1 is deemed to be non trivial. In this paper we refine the technique of enhanced negative type and show how it may be applied more generally to any finite metric space (X,d) that is known to have strict p-negative type for some non negative p. This allows us to significantly improve the lower bounds on the supremal strict p-negative type of finite metric trees that were given by Doust and Weston and, moreover, leads in to one of our main results: The supremal p-negative type of a finite metric space cannot be strict. By way of application we are then able to exhibit large classes of finite metric spaces (such as finite isometric subspaces of Hadamard manifolds) that must have strict p-negative type for some p > 1. We also show that if a metric space (finite or otherwise) has p-negative type for some p > 0, then it must have strict q-negative type for all q in [0,p). This generalizes a well known theorem of Schoenberg and leads to a complete classification of the intervals on which a metric space may have strict p-negative type. (Several of the results in this paper hold more generally for semi-metric spaces.)