On the magnitude function of domains in Euclidean space

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 ${\mathcal{M}}_X(R)=\mathrm{m}\mathrm{a}\mathrm{g}(R\cdot X)$. 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 $R\to \mathrm{\infty }$, ${\mathcal{M}}_X$ admits an asymptotic expansion. The three leading terms of ${\mathcal{M}}_X$ at $R=+\mathrm{\infty }$ 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.