Spread: a measure of the size of metric spaces

TL;DR

The paper introduces the concept of spread as a new measure of the size of metric spaces, connecting it to existing notions like magnitude and exploring its properties across finite, infinite, and Riemannian spaces.

Contribution

It defines spread for finite and infinite metric spaces, relates it to magnitude, and introduces a scale-dependent dimension with applications to fractals and Riemannian manifolds.

Findings

Spread relates to magnitude and volume in metric spaces.

The scale-dependent dimension approximates Hausdorff dimension.

Spread can be computed for spheres, lines, and certain fractals.

Abstract

Motivated by Leinster-Cobbold measures of biodiversity, the notion of the spread of a finite metric space is introduced. This is related to Leinster's magnitude of a metric space. Spread is generalized to infinite metric spaces equipped with a measure and is calculated for spheres and straight lines. For Riemannian manifolds the spread is related to the volume and total scalar curvature. A notion of scale-dependent dimension is introduced and seen, numerically, to be close to the Hausdorff dimension for approximations to certain fractals.