# Positive definite metric spaces

Positive definite metric spaces Magnitude is a numerical invariant of finite metric spaces, recently introduced by 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 $$\ell _p^n$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Positive definite metric spaces

, Volume 17 (3) – Sep 16, 2012
25 pages

/lp/springer_journal/positive-definite-metric-spaces-ZsJcAeFQx5
Publisher
Springer Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-012-0202-8
Publisher site
See Article on Publisher Site

### Abstract

Magnitude is a numerical invariant of finite metric spaces, recently introduced by 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 $$\ell _p^n$$ 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.

### Journal

PositivitySpringer Journals

Published: Sep 16, 2012

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