# Positive definite metric spaces

@article{Meckes2010PositiveDM, title={Positive definite metric spaces}, author={Mark W. Meckes}, journal={Positivity}, year={2010}, volume={17}, pages={733-757} }

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…

## 48 Citations

### On the magnitudes of compact sets in Euclidean spaces

- Mathematics
- 2015

abstract:The notion of the magnitude of a metric space was introduced by Leinster and developed in works by Leinster, Meckes and Willerton, but the magnitudes of familiar sets in Euclidean space are…

### The magnitude of metric spaces

- MathematicsDocumenta Mathematica
- 2013

Magnitude is a real-valued invariant of metric spaces, analogous to the Euler characteristic of topological spaces and the cardinality of sets. The definition of magnitude is a special case of a…

### ON THE MAGNITUDE AND INTRINSIC VOLUMES OF A CONVEX BODY IN EUCLIDEAN SPACE

- MathematicsMathematika
- 2020

Magnitude is an isometric invariant of metric spaces inspired by category theory. Recent work has shown that the asymptotic behavior under rescaling of the magnitude of subsets of Euclidean space is…

### Approximating the Convex Hull via Metric Space Magnitude

- Mathematics, Computer Science
- 2019

This paper restricts the sets to finite subsets of Euclidean space and investigates its individual components, and gives an explicit formula for the corrected inclusion-exclusion principle, and defines a quantity associated with each point, called the moment, which gives an intrinsic ordering to the points.

### On the magnitude of odd balls via potential functions

- Mathematics
- 2018

Magnitude is a measure of size defined for certain classes of metric spaces; it arose from ideas in category theory. In particular, magnitude is defined for compact subsets of Euclidean space and, in…

### THE MAGNITUDE OF A METRIC SPACE: FROM CATEGORY THEORY TO GEOMETRIC MEASURE THEORY

- Mathematics
- 2017

Magnitude is a numerical isometric invariant of metric spaces, whose definition arises from a precise analogy between categories and metric spaces. Despite this exotic provenance, magnitude turns out…

### On the magnitude of a finite dimensional algebra

- Mathematics
- 2015

There is a general notion of the magnitude of an enriched category, defined subject to hypotheses. In topological and geometric contexts, magnitude is already known to be closely related to classical…

### The magnitude and spectral geometry

- Mathematics
- 2022

We study the geometric significance of Leinster’s notion of magnitude for a smooth manifold with boundary of arbitrary dimension, motivated by open questions for the unit disk in R2. For a large…

### The maximum entropy of a metric space

- Mathematics, Computer ScienceArXiv
- 2019

We define a one-parameter family of entropies, each assigning a real number to any probability measure on a compact metric space (or, more generally, a compact Hausdorff space with a notion of…

### On the asymptotic magnitude of subsets of Euclidean space

- MathematicsGeometriae Dedicata
- 2012

Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of…

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