Magnitude, diversity, capacities, and dimensions of metric spaces

TL;DR

This paper extends the concept of magnitude from finite to compact metric spaces, revealing a new link between magnitude and capacities, and demonstrating that magnitude dimension equals Minkowski dimension in Euclidean spaces.

Contribution

It introduces a more natural definition of magnitude for compact spaces and establishes a novel relationship between magnitude and capacities.

Findings

Magnitude for compact spaces aligns with previous finite definitions

Magnitude dimension equals Minkowski dimension in Euclidean spaces

New relationship between magnitude and capacities of sets

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.