On the magnitudes of compact sets in Euclidean spaces

J. A. Barcelo, Anthony Carbery

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

The notion of the magnitude of a metric space was introduced
by Leinster in [11] and developed in [16], [12], [17], [20] and [13], but the magnitudes of familiar sets in Euclidean space are only understood in relatively few cases. In this paper we study the magnitudes of compact sets in Euclidean spaces. We first describe the asymptotics of the magnitude of such sets in both the small and large-scale regimes. We then consider the magnitudes of compact convex sets with nonempty interior in Euclidean spaces of odd dimension, and relate them to the boundary behaviour of solutions to certain naturally associated higher order elliptic boundary value problems in exterior domains. We carry out calculations leading to an algorithm for explicit evaluation of the magnitudes of balls, and this establishes the convex magnitude conjecture of Leinster and Willerton [12] in the special case of balls in dimension three. In general we show that the magnitude of an odd-dimensional ball is a rational function of its radius. In addition to Fourier-analytic and PDE techniques, the arguments also involve some combinatorial considerations.
Original languageEnglish
Pages (from-to)449-494
Number of pages39
JournalAmerican Journal of Mathematics
Issue number2
Early online date16 Mar 2018
Publication statusPublished - Apr 2018

Keywords / Materials (for Non-textual outputs)

  • 51F99 Metric Geometry
  • 42B99 Harmonic Analysis in Several Variables


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