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

Functional Analysis;Analysis of PDEs