abstract: The search for second order sufficient minimality conditions is a very well studied issue in the Calculus of Variations. Nevertheless, in the framework of free-discontinuity problems, this kind of study has been started only in recent years. In this talk, I will present a second variation approach to a nonlocal variant of the isoperimetric problem, leading to some new local and, in some cases, global minimality results. This is joint work with Marco Bonacini.