Abstract
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 halfline 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.
Original language  English 

Pages (fromto)  939967 
Number of pages  29 
Journal  American Journal of Mathematics 
Volume  143 
Issue number  3 
DOIs  
Publication status  Published  8 Jun 2021 
ASJC Scopus subject areas
 Mathematics(all)
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Heiko Gimperlein
 School of Mathematical & Computer Sciences  Associate Professor
 School of Mathematical & Computer Sciences, Mathematics  Associate Professor
Person: Academic (Research & Teaching)