Spaces of extremal magnitude

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.

7 pages

References in corpus (3)

• The magnitude of a metric space: from category theory to geometric measure theory
• On the magnitude function of domains in Euclidean space
• The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials