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 domainXXin an odd-dimensional Euclidean space, we show that the asymptotic expansion of the functionMX(R)=Mag(R⋅X)\mathcal {M}_X(R) = \mathrm {Mag}(R\cdot X)atR=∞R = \inftydetermines the Willmore energy of the boundary∂X\partial X. This disproves the Leinster-Willerton conjecture for a compact convex body in odd dimensions.