The magnitude of metric spaces
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 , 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. 857–905DOI 10.4171/DM/415