Maz’ya–Shaposhnikova Meet Bishop–Gromov

We find a surprising link between Maz’ya–Shaposhnikova’s well-known asymptotic formula concerning fractional Sobolev seminorms and the generalized Bishop–Gromov inequality. In the setting of abstract metric measure spaces we prove the validity of a large family of asymptotic formulas concerning non-local energies. Important examples which are covered by our approach are for instance Carnot groups, Riemannian manifolds with Ricci curvature bounded from below and non-collapsed RCD spaces. We also extend the classical Maz’ya–Shaposhnikova’s formula on Euclidean spaces to a wider class of mollifiers.