# The magnitude of metric spaces

### Tom Leinster

## Abstract

Magnitude is a real-valued invariant of metric spaces, analogous to Euler characteristic of topological spaces and cardinality of sets. The definition of magnitude is a special case of a general categorical definition that clarifies the analogies between 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 joint work with Willerton) that magnitude encodes all the most important invariants of classical integral geometry.

## Cite this article

Tom Leinster, The magnitude of metric spaces. Doc. Math. 18 (2013), pp. 857–905

DOI 10.4171/DM/415