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arXiv · 1209.2300

Spread: a measure of the size of metric spaces

Simon Willerton
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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.

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Simon Willerton. 2015-01-06. Spread: a measure of the size of metric spaces. https://arxiv.org/abs/1209.2300

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