Title: On the magnitude function of domains in Euclidean space,
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.