The Willmore energy and the magnitude of Euclidean domains

TL;DR

This paper links the asymptotic behavior of the magnitude function of smooth, compact domains in odd-dimensional Euclidean spaces to the Willmore energy of their boundaries, challenging previous conjectures.

Contribution

It establishes a new connection between magnitude and Willmore energy, disproving the Leinster-Willerton conjecture in all odd dimensions for convex bodies.

Findings

Asymptotic expansion of magnitude determines boundary Willmore energy

Disproves Leinster-Willerton conjecture for convex bodies in odd dimensions

Provides new insights into geometric significance of magnitude

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.