Positive definite metric spaces

TL;DR

This paper introduces positive definite metric spaces, a class where the magnitude invariant is more manageable, and explores their properties and implications for compact subsets of l_p^n spaces.

Contribution

It develops the theory of positive definite metric spaces, extending the concept of magnitude and proving its equivalence across definitions for these spaces.

Findings

All definitions of magnitude coincide for compact positive definite metric spaces.

Magnitude behaves predictably as a function of such spaces.

Results extend Leinster’s work to compact subsets of l_p^n for p ≤ 2.

Abstract

Magnitude is a numerical invariant of finite metric spaces, recently introduced by T. Leinster, which is analogous in precise senses to the cardinality of finite sets or the Euler characteristic of topological spaces. It has been extended to infinite metric spaces in several a priori distinct ways. This paper develops the theory of a class of metric spaces, positive definite metric spaces, for which magnitude is more tractable than in general. Positive definiteness is a generalization of the classical property of negative type for a metric space, which is known to hold for many interesting classes of spaces. It is proved that all the proposed definitions of magnitude coincide for compact positive definite metric spaces and further results are proved about the behavior of magnitude as a function of such spaces. Finally, some facts about the magnitude of compact subsets of l_p^n for p \le 2 are proved, generalizing results of Leinster for p=1,2, using properties of these spaces which are somewhat stronger than positive definiteness.