The Willmore energy and the magnitude of Euclidean domains

Abstract

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.