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

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