  We study the geometric significance of Leinster's notion of magnitude for a
compact metric space. For a smooth, compact domain in an odd-dimensional
Euclidean space, we show that the asymptotic expansion of the magnitude
function at infinity determines the Willmore energy of the boundary. This
disproves the Leinster-Willerton conjecture for a compact convex body in all
odd dimensions.

Metric Geometry;Analysis of PDEs;Classical Analysis and ODEs;Differential Geometry