  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.
