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… 
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23 Citations
Highly Influential Citations
3
Background Citations
14
Methods Citations
2

23 Citations

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Approximating the Convex Hull via Metric Space Magnitude
TLDR
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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…
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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…
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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…
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References

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TLDR
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