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.

20 pages, 3 figures, to appear in American Journal of Mathematics

References in corpus (1)

• On the magnitude function of domains in Euclidean space

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