# Magnitude, Diversity, Capacities, and Dimensions of Metric Spaces

@article{Meckes2013MagnitudeDC, title={Magnitude, Diversity, Capacities, and Dimensions of Metric Spaces}, author={Mark W. Meckes}, journal={Potential Analysis}, year={2013}, volume={42}, pages={549-572} }

Magnitude is a numerical invariant of metric spaces introduced by Leinster, motivated by considerations from category theory. This paper extends the original definition for finite spaces to compact spaces, in an equivalent but more natural and direct manner than in previous works by Leinster, Willerton, and the author. The new definition uncovers a previously unknown relationship between magnitude and capacities of sets. Exploiting this relationship, it is shown that for a compact subset of…

## 27 Citations

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…

Spaces of extremal magnitude

- Mathematics
- 2021

Magnitude is a numerical invariant of compact metric spaces. Its theory is most mature for spaces satisfying the classical condition of being of negative type, and the magnitude of such a space lies…

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…

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…

Spread: A Measure of the Size of Metric Spaces

- MathematicsInt. J. Comput. Geom. Appl.
- 2015

A notion of scale-dependent dimension is introduced and seen for approximations to certain fractals to be numerically close to the Minkowski dimension of the original fractals.

Magnitude homology of enriched categories and metric spaces

- MathematicsAlgebraic & Geometric Topology
- 2021

Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as $[0,\infty)$-enriched categories. We show that in many cases magnitude can be categorified to 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…

The Willmore energy and the magnitude of Euclidean domains

- Mathematics
- 2021

We study the geometric significance of Leinster’s notion of magnitude for a compact metric space. For a smooth, compact domain X in an odd-dimensional Euclidean space, we show that the asymptotic…

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…

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…

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