Local and global minimality results for a nonlocal isoperimetric problem on R^N

We consider a nonlocal isoperimetric problem defined in the whole space ${\mathbb{R}}^N$, 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.