LOCAL AND GLOBAL MINIMALITY RESULTS FOR A NONLOCAL ISOPERIMETRIC PROBLEM ON RN

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 is an element of(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 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.