The magnitude of metric spaces

Abstract

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.

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

References in corpus (6)

• Heuristic and computer calculations for the magnitude of metric spaces
• Euler Characteristics of Categories and Homotopy Colimits
• Euler characteristics of p-subgroup categories
• Notions of Möbius inversion
• The zeta function of a finite category which has Möbius inversion
• The zeta function of a finite category and the series Euler characteristic

Cited by in corpus (23)

• Categorifying the magnitude of a graph
• The magnitude of a metric space: from category theory to geometric measure theory
• Tight spans, Isbell completions and semi-tropical modules
• On the magnitude function of domains in Euclidean space
• Magnitude meets persistence. Homology theories for filtered simplicial sets
• Persistent Magnitude
• Graph magnitude homology via algebraic Morse theory
• Magnitude homology of enriched categories and metric spaces
• The Legendre-Fenchel transform from a category theoretic perspective
• The Willmore energy and the magnitude of Euclidean domains
• Entropy and Diversity: The Axiomatic Approach
• On the magnitude of odd balls via potential functions
• Girth, magnitude homology, and phase transition of diagonality
• Spread: a measure of the size of metric spaces
• Smoothness filtration of the magnitude complex
• The magnitude vector of images
• Geometric approach to graph magnitude homology
• Euler characteristic for weak n-categories and $\left(\infty ,1\right)$-categories
• The Euler characteristic of a bicategory and the product formula for fibered bicategories
• The Euler characteristic of an enriched category
• Practical applications of metric space magnitude and weighting vectors
• Approximating the Convex Hull via Metric Space Magnitude
• Magnitude homology of graphs and discrete Morse theory on Asao-Izumihara complexes