Leinster, Tom

The magnitude of metric spaces

Doc. Math., J. DMV 18, 857-905 (2013)


Summary: 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.

Mathematics Subject Classification

51F99, 18D20, 18F99, 28A75, 49Q20, 52A20, 52A38, 53C65


metric space, magnitude, enriched category, Möbius inversion, Euler characteristic of a category, finite metric space, convex set, integral geometry, valuation, intrinsic volume, fractal dimension, positive definite space, space of negative type