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 $\{R}^n$, 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.

2010 Mathematics Subject Classification: 51F99 (primary), 18D20, 18F99, 28A75, 49Q20, 52A20, 52A38, 53C65 (secondary).

Keywords and Phrases: 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.

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