On the asymptotic magnitude of subsets of Euclidean space

TL;DR

This paper investigates the asymptotic behavior of the magnitude invariant for subsets of Euclidean space, connecting it to intrinsic volumes and the inclusion-exclusion principle through approximation methods.

Contribution

It introduces a method to approximate the magnitude of compact Euclidean subsets and establishes their asymptotic relation to intrinsic volumes and inclusion-exclusion.

Findings

Magnitudes of line segments, circles, and Cantor sets are computed.

Asymptotically, these magnitudes satisfy the inclusion-exclusion principle.

Results relate magnitude to intrinsic volumes of polyconvex sets.

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.