Local and global minimality results for a nonlocal isoperimetric problem on RN

We consider a nonlocal isoperimetric problem defined in the whole space RN , whose nonlocal part is given by a Riesz potential with exponent α ⋯ (0,N - 1). We show that critical configurations with positive second variation are local minimizers and satisfy a quantitative inequality with respect to the L1 -norm. This criterion provides the existence of an (explicitly determined) critical threshold determining the interval of volumes for which the ball is a local minimizer, and it allows us to address several global minimality issues.