Abstract

abstract:

We study Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain $X\subset\Bbb{R}^{2m-1}$, we find geometric significance in the function $\scr{M}_X(R)={\rm mag}(R\cdot X)$. The function $\scr{M}_X$ extends from the positive half-line to a meromorphic function in the complex plane. Its poles are generalized scattering resonances. In the semiclassical limit $R\to\infty$, $\scr{M}_X$ admits an asymptotic expansion. The three leading terms of $\scr{M}_X$ 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.

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