  A symmetrization techique, introduced by Cianchi, Fusco, Maggi and Pratelli
concerning the Sobolev inequality, is adapted to the
Gagliardo-Nirenberg-Sobolev inequality (GNS) to obtain a reduction step of the
problem of showing its quantitative version. More precisely we prove a
stability result for the GNS inequality under the hypothesis that it holds, in
turn, in the smaller class of radial symmetric decreasing functions.
