THE WILLMORE ENERGY AND THE MAGNITUDE OF EUCLIDEAN DOMAINS
We study the geometric significance of Leinster’s notion of magnitude for a compact metric space. For a smooth, compact domain X in an odd-dimensional Euclidean space, we show that the asymptotic expansion of the function MX(R) = Mag(R·X) at R = ∞ determines the Willmore energy of the boundary ∂X. This disproves the Leinster-Willerton conjecture for a compact convex body in odd dimensions.