The magnitude of metric spaces

Tom Leinster

Documenta mathematica, Tome 18 (2013), pp. 857-905

Résumé

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 Rn, 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.

Détail

DOI : 10.4171/dm/415

Classification : 18D20, 18F99, 28A75, 49Q20, 51F99, 52A20, 52A38, 53C65
Mots-clés : metric space, fractal dimension, valuation, Möbius inversion, Euler characteristic of a category, magnitude, enriched category, finite metric space, convex set, integral geometry, intrinsic volume, positive definite space, space of negative type