The magnitude of metric spaces

Details The magnitude of metric spaces The magnitude of metric spaces

2010-12-29

arxiv.org/abs/1012.5857v3

Mathematics

Metric Geometry

35 pages. v3: added discussion of negative type; formatting changed; minor edits

Scientific paper

Magnitude is a real-valued invariant of metric spaces, analogous to the Euler characteristic of topological spaces and the cardinality of sets. The definition of magnitude is a special case of a general categorical definition that clarifies the analogies between various 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 arXiv:0908.1582) that magnitude subsumes all the most important invariants of classical integral geometry.

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