Spaces of extremal magnitude

TL;DR

This paper constructs the first known examples of compact metric spaces with infinite magnitude, answering an open question since 2010, and provides conditions for magnitude convergence to 1 under scaling.

Contribution

It introduces the first examples of spaces with infinite magnitude and unifies conditions for magnitude convergence to 1 as spaces shrink.

Findings

Constructed examples of spaces with infinite magnitude

Provided a sufficient condition for magnitude to approach 1 upon scaling

Unified previous results on magnitude convergence

Abstract

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 in the interval $[1, \infty]$. Until now, no example with magnitude $\infty$ was known. We construct some, thus answering a question open since 2010. We also give a sufficient condition for the magnitude of a space to converge to 1 as it is scaled down to a point, unifying and generalizing previously known conditions.