On the asymptotic magnitude of subsets of Euclidean space

By Tom Leinster, Simon Willerton. 2009

48 citations recorded by the source corpus.

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 Euclidean space with finite subsets, the magnitudes of line segments, circles and Cantor sets are defined and calculated. It is observed that asymptotically these satisfy the inclusion-exclusion principle, relating them to intrinsic volumes of polyconvex sets.

Subfields: Computer Vision and Pattern Recognition, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a839353f56ac8199d21218b01c779cf9de2dcd82

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Citations in the corpus

Magnitude, Diversity, Capacities, and Dimensions of Metric Spaces (2013, background): “In addition, the conjectures from [9, 11] mentioned earlier would, if true, indicate previously unknown connections between capacities and intrinsic volumes of convex bodies.”
Positive definite metric spaces (2010, background): “mma 2.8 and Corollary 2.10 below.) In [42], Willerton showed that the magnitudes of intervals, circles, and Cantor sets, as computed via weight measures, coincide with their magnitudes as computed in [25] using the second approach. The results of this section show that all these approaches to deﬁnin”
Maximizing diversity in biology and beyond (2015, unknown): “By definition, the magnitude of a finite metric space X is the magnitude of the matrix Z = (e)i,j∈X ; see [13,38,45], for instance.”
The magnitude of metric spaces (2010, background): “In the terminology of statistical mechanics, the denominator is the partition function for the energies Ei at inverse temperature t. 4 4I thank Simon Willerton for suggesting that some such relationship should exist. iii.”
The magnitude of a graph (2014, background): “I am especially grateful to David Speyer and Simon Willerton for sharing their thoughts, and for allowing me to include Proposition 5.1.”
Categorifying the magnitude of a graph (2015, background): “This was found to have connections with topics as varied as intrinsic volumes [13], biodiversity [12], potential theory [16], Minkowski dimension [16] and curvature [20].”