# On the magnitude function of domains in Euclidean space

## TL;DR

This paper explores the geometric interpretation of the magnitude function for smooth, compact domains in Euclidean space, revealing its connection to scattering resonances and classical geometric measures like volume and curvature.

## Contribution

It establishes the asymptotic expansion of the magnitude function for domains in Euclidean space, linking it to intrinsic volumes and correcting previous conjectures.

## Key findings

- The magnitude function extends meromorphically with poles as scattering resonances.
- Asymptotic expansion at infinity relates to volume, surface area, and mean curvature.
- For convex domains, the expansion confirms and refines the convex magnitude conjecture.

## Abstract

We study Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain $X\subset \mathbb{R}^{2m-1}$, we find geometric significance in the function $\mathcal{M}_X(R) = \mathrm{mag}(R\cdot X)$. The function $\mathcal{M}_X$ extends from the positive half-line to a meromorphic function in the complex plane. Its poles are generalized scattering resonances. In the semiclassical limit $R \to \infty$, $\mathcal{M}_X$ admits an asymptotic expansion. The three leading terms of $\mathcal{M}_X$ at $R=+\infty$ are proportional to the volume, surface area and integral of the mean curvature. In particular, for convex $X$ the leading terms are proportional to the intrinsic volumes, and we obtain an asymptotic variant of the convex magnitude conjecture by Leinster and Willerton, with corrected coefficients.