Magnitude, Diversity, Capacities, and Dimensions of Metric Spaces
By Mark W. Meckes. 2013
58 citations recorded by the source corpus.
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 spaces, in an equivalent but more natural and direct manner than in previous works by Leinster, Willerton, and the author. The new definition uncovers a previously unknown relationship between magnitude and capacities of sets. Exploiting this relationship, it is shown that for a compact subset of Euclidean space, the magnitude dimension considered by Leinster and Willerton is equal to the Minkowski dimension.
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/08d5bdf9787bb5d1c4657cc7722f9eb35e76f12e
How researchers see this paper
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In Computational Theory and Mathematics
In Geometry and Topology
- Michael Shulman
- Richard Hepworth
- Simon Willerton
- Tom Leinster
- Mark W. Meckes
- Dominic R. Verity
- Emily Riehl
- R. Haugseng
In Mathematical Physics
- Richard Hepworth
- Simon Willerton
- Michael Shulman
- Tom Leinster
- Mark W. Meckes
- Denis-Charles Cisinski
- Dominic R. Verity
- Emily Riehl
Citations in the corpus
- Categorifying the magnitude of a graph (2015, background): “2 of [16] — but certainly all computed examples are increasing.”
- Magnitude homology of enriched categories and metric spaces (2017, background): “V support an interesting magnitude homology? An interesting possibility is the category ([0,∞],≥) with the (cartesian) monoidal structure max, whose enriched categories are ultrametric spaces; Meckes [Mec15] has shown that the magnitude of ultrametric spaces detects their entropy/capacity. (3) In th”
- The magnitude of a graph (2014, background): “area, volume, and so on. The magnitude of metric spaces is also closely related to certain measures of entropy and of biological diversity [7], and admits a further potential-theoretic interpretation [13]. Graphs are metric spaces, with distance between vertices measured as the length of a shortest ”