ON THE MAGNITUDE FUNCTION OF DOMAINS IN EUCLIDEAN SPACE

Journal article, 2021

We study Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain X subset of R2m-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 -> infinity, M-X admits an asymptotic expansion. The three leading terms of M-X at R = +infinity 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.