THE MAGNITUDE OF A METRIC SPACE: FROM CATEGORY THEORY TO GEOMETRIC MEASURE THEORY

@article{Leinster2017THEMO,
  title={THE MAGNITUDE OF A METRIC SPACE: FROM CATEGORY THEORY TO GEOMETRIC MEASURE THEORY},
  author={Tom Leinster and Mark W. Meckes},
  journal={arXiv: Metric Geometry},
  year={2017},
  pages={156-193}
}
Magnitude is a numerical isometric invariant of metric spaces, whose definition arises from a precise analogy between categories and metric spaces. Despite this exotic provenance, magnitude turns out to encode many invariants from integral geometry and geometric measure theory, including volume, capacity, dimension, and intrinsic volumes. This paper gives an overview of the theory of magnitude, from its category-theoretic genesis to its connections with these geometric quantities. Some new… 
Magnitude homology of enriched categories and metric spaces
Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as $[0,\infty)$-enriched categories. We show that in many cases magnitude can be categorified to a
ON THE MAGNITUDE AND INTRINSIC VOLUMES OF A CONVEX BODY IN EUCLIDEAN SPACE
Magnitude is an isometric invariant of metric spaces inspired by category theory. Recent work has shown that the asymptotic behavior under rescaling of the magnitude of subsets of Euclidean space is
Approximating the Convex Hull via Metric Space Magnitude
TLDR
This paper restricts the sets to finite subsets of Euclidean space and investigates its individual components, and gives an explicit formula for the corrected inclusion-exclusion principle, and defines a quantity associated with each point, called the moment, which gives an intrinsic ordering to the points.
Magnitude and Holmes-Thompson intrinsic volumes of convex bodies
. Magnitude is a numerical invariant of compact metric spaces, originally in-spired by category theory and now known to be related to myriad other geometric quanti-ties. Generalizing earlier results
Spaces of extremal magnitude
Magnitude is a numerical invariant of compact metric spaces. Its theory is most mature for spaces satisfying the classical condition of being of negative type, and the magnitude of such a space lies
The magnitude and spectral geometry
We study the geometric significance of Leinster’s notion of magnitude for a smooth manifold with boundary of arbitrary dimension, motivated by open questions for the unit disk in R2. For a large
Magnitude meets persistence. Homology theories for filtered simplicial sets
The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely
On the magnitude of odd balls via potential functions
Magnitude is a measure of size defined for certain classes of metric spaces; it arose from ideas in category theory. In particular, magnitude is defined for compact subsets of Euclidean space and, in
Alpha magnitude
Magnitude is an isometric invariant for metric spaces that was introduced by Leinster around 2010, and is currently the object of intense research, since it has been shown to encode many known
The Willmore energy and the magnitude of Euclidean domains
. We study the geometric significance of Leinster’s notion of magnitude for a compact metric space. For a smooth, compact domain X in an odd-dimensional Euclidean space, we show that the asymptotic
...
...

References

SHOWING 1-10 OF 42 REFERENCES
The magnitude of metric spaces
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
On the asymptotic magnitude of subsets of Euclidean space
Magnitude is a canonical invariant of finite metric spaces which has its origins in category theory; it is analogous to cardinality of finite sets. Here, by approximating certain compact subsets of
Does negative type characterize the round sphere
We discuss the measure-theoretic metric invariants extent, mean distance and symmetry ratio and their relation to the concept of negative type of a metric space. A conjecture stating that a compact
On the magnitude of a finite dimensional algebra
There is a general notion of the magnitude of an enriched category, defined subject to hypotheses. In topological and geometric contexts, magnitude is already known to be closely related to classical
On the magnitude of spheres, surfaces and other homogeneous spaces
In this paper we calculate the magnitude of metric spaces using measures rather than finite subsets as had been done previously. An explicit formula for the magnitude of an $$n$$-sphere with its
Heuristic and computer calculations for the magnitude of metric spaces
The notion of the magnitude of a compact metric space was considered in arXiv:0908.1582 with Tom Leinster, where the magnitude was calculated for line segments, circles and Cantor sets. In this paper
Magnitude, Diversity, Capacities, and Dimensions of Metric Spaces
Magnitude is a numerical invariant of metric spaces introduced by Leinster, motivated by considerations from category theory. This paper extends the original definition for finite spaces to compact
Positive definite metric spaces
Magnitude is a numerical invariant of finite metric spaces, recently introduced by Leinster, which is analogous in precise senses to the cardinality of finite sets or the Euler characteristic of
Metric Structures for Riemannian and Non-Riemannian Spaces
Length Structures: Path Metric Spaces.- Degree and Dilatation.- Metric Structures on Families of Metric Spaces.- Convergence and Concentration of Metrics and Measures.- Loewner Rediscovered.-
Finite metric spaces: combinatorics, geometry and algorithms
TLDR
In this talk the metric theory of finite graphs has proved very useful in the study of graphs per se and the design of approximation algorithms for hard computational problems.
...
...