  We consider a nonlocal isoperimetric problem defined in the whole space
$\R^N$, whose nonlocal part is given by a Riesz potential with exponent
$\alpha\in(0, N-1)$. We show that critical configurations with positive second
variation are local minimizers and satisfy a quantitative inequality with
respect to the $L^1$-norm. This criterion provides the existence of a
(explicitly determined) critical threshold determining the interval of volumes
for which the ball is a local minimizer, and allows to address several global
minimality issues.
