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 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.