A spherical rearrangement proof of the stability of a Riesz-type inequality and an application to an isoperimetric type problem

We prove the stability of ball as global minimizer an attractive shape functional under volume constraint, by means mass transportation arguments. The exponent is $1/2$ and it sharp. Moreover, we use such result together with quantitative (possibly fractional) isoperimetric inequality to that a involving both repulsive term sufficiently large fixed suitable perimeter penalization.