Magnitude, diversity, capacities, and dimensions of metric spaces

2013/08/31 by Mark W. Meckes · 1 citation
Mathematics · #math.MG #math.CA #math.FA

paper · pdf · doi:10.1007/s11118-014-9444-3

published as Potential Anal. 42 (2015) no. 2, 549-572 · v2: Minor changes in exposition. To appear in Potential Analysis

arxiv created 2014/09/30 · arxiv updated 2015/07/22

Abstract

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 Euclidean space, the magnitude dimension considered by Leinster and Willerton is equal to the Minkowski dimension.

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