On the asymptotic magnitude of subsets of Euclidean space

Abstract

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.Comment: 23 pages. Version 2: updated to reflect more recent work, in particular, the approximation method is now known to calculate (rather than merely define) the magnitude; also minor alterations such as references adde

Categorifying the magnitude of a graph

“…The magnitude of a finite metric space was introduced by Leinster [6] by analogy with his notion of the Euler characteristic of a category [5]. This was found to have connections with topics as varied as intrinsic volumes [10], biodiversity [9], potential theory [12], Minkowski dimension [12] and curvature [14].…”

Maximizing Diversity in Biology and Beyond

“…This has been most intensively studied in the context of metric spaces. By definition, the magnitude of a finite metric space X is the magnitude of the matrix Z = (e −d(i,j) ) i,j∈X ; see [24,27,30], for instance. In the metric context, the meaning of magnitude becomes clearer after one extends the definition from finite to compact spaces (which is done by approximating them by finite subspaces).…”

The magnitude of a metric space: from category theory to geometric measure theory

“…Categories are a special case of a more general family of structures, enriched categories, which encompass both categories with additional structure (like linear categories) and, surprisingly, metric spaces. In [23,19], the definition of Euler characteristic of a category was generalized to enriched categories, renamed magnitude, then re-specialized to finite metric spaces. The first paper to be written on magnitude [23] focused on the asymptotic behavior of the magnitudes of finite approximations to specific compact subsets of Euclidean space.…”

“…In [23,19], the definition of Euler characteristic of a category was generalized to enriched categories, renamed magnitude, then re-specialized to finite metric spaces. The first paper to be written on magnitude [23] focused on the asymptotic behavior of the magnitudes of finite approximations to specific compact subsets of Euclidean space. The results there hinted strongly that magnitude is closely related to geometric quantities including volume and fractal dimension; numerical computations in [40] gave further evidence of these relationships.…”

“…In [41], a definition was proposed for the magnitude of certain compact metric spaces, and connections were found between magnitude and some intrinsic volumes of Riemannian manifolds. Shortly thereafter, the paper [19] appeared which laid out for the first time the general theory of the magnitude of finite metric spaces; and [27] which put the asymptotic approach of [23] for studying magnitude of compact spaces on firm footing, and showed that it also coincides with the definition used in [41].…”