# 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}
}
• M. Meckes
• Published 25 August 2013
• Mathematics
• Potential Analysis
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…
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## References

SHOWING 1-10 OF 22 REFERENCES
The magnitude of metric spaces
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
Heuristic and computer calculations for the magnitude of metric spaces
The notion of the magnitude of a compact metric space was considered in arXiv:0908.1582 with Tom Leinster, where the magnitude was calculated for line segments, circles and Cantor sets. In this paper
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
On the asymptotic magnitude of subsets of Euclidean space
• Mathematics
• 2013
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
Asymptotic Theory Of Finite Dimensional Normed Spaces
• Mathematics
• 1986
The Concentration of Measure Phenomenon in the Theory of Normed Spaces.- Preliminaries.- The Isoperimetric Inequality on Sn?1 and Some Consequences.- Finite Dimensional Normed Spaces, Preliminaries.-
On the magnitude of spheres, surfaces and other homogeneous spaces
In this paper we calculate the magnitude of metric spaces using measures rather than finite subsets as had been done previously. An explicit formula for the magnitude of an $$n$$-sphere with its
Function Spaces and Potential Theory
• Mathematics
• 1995
The subject of this book is the interplay between function space theory and potential theory. A crucial step in classical potential theory is the identification of the potential energy of a charge
Introduction to Fourier Analysis on Euclidean Spaces.
• Mathematics
• 1971
The authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action
A maximum entropy theorem with applications to the measurement of biodiversity
This is a preliminary article stating and proving a new maximum entropy theorem. The entropies that we consider can be used as measures of biodiversity. In that context, the question is: for a given
Metric Spaces
• Mathematics
• 1990
The papers [3], [7], [2], [1], [5], [6], and [4] provide the notation and terminology for this paper. We consider metric structures which are systems 〈 a carrier, a distance 〉 where the carrier is a