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.

PB - SIAM Publications VL - 46 UR - http://hdl.handle.net/1963/6984 IS - 4 U1 - 6976 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER -