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 ⊂ R ^2m−1, we find geometric significance in the function M _X (R) =mag(R · X). The function 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 → ∞, M _X admits an asymptotic expansion. The three leading terms of M _X at R =+∞ 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.