On the magnitudes of compact sets in Euclidean spaces

TL;DR

This paper investigates the magnitude of compact sets in Euclidean spaces, providing asymptotic descriptions, explicit calculations for balls, and confirming the convex magnitude conjecture in three dimensions.

Contribution

It introduces new methods to compute magnitudes of Euclidean sets, especially convex bodies, and proves the convex magnitude conjecture for three-dimensional balls.

Findings

Magnitude of odd-dimensional balls is a rational function of radius.

Established asymptotic behavior of magnitudes at small and large scales.

Confirmed the convex magnitude conjecture for three-dimensional balls.

Abstract

The notion of the magnitude of a metric space was introduced by Leinster in [8] and developed in [10], [9], [11] and [16], 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 [9] 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.