The magnitude of metric spaces

TL;DR

This paper explores the concept of magnitude as a metric space invariant, revealing its deep connections to classical geometric measures and proposing it as a unifying invariant in integral geometry.

Contribution

It introduces magnitude as a categorical invariant of metric spaces and demonstrates its relationship to classical geometric measures, suggesting it subsumes key invariants in integral geometry.

Findings

Magnitude relates to volume, surface area, and dimension in R^n

Evidence supports the conjecture that magnitude unifies classical invariants

Magnitude generalizes Euler characteristic and set cardinality

Abstract

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 general categorical definition that clarifies the analogies between various cardinality-like invariants in mathematics. Although this motivation is a world away from geometric measure, magnitude, when applied to subsets of R^n, turns out to be intimately related to invariants such as volume, surface area, perimeter and dimension. We describe several aspects of this relationship, providing evidence for a conjecture (first stated in arXiv:0908.1582) that magnitude subsumes all the most important invariants of classical integral geometry.