Abstract: Given a convex body containing the origin, one can consider the norm for which is the unit ball (which is in fact not precisely a norm since is not necessarily symmetric with respect to the origin). Using the standard dual norm to weigh the lengths of the unit vectors, this leads to the well-known notion of anisotropic perimeter. As for the Euclidean case, one can show the anisotropic isoperimetric inequality, stating that among all bodies with a fixed volume, the least perimeter is reached in the case when E coincides with the set itself up to translation and rescaling. A quantitative version of this inequality must relate the deficit in this inequality with the distance, in some sense, between the set E and (a rescaled-translated of) . Both the isoperimetric inequality and this quantitative version have been well studied in the years, giving rise to many interesting results. In this talk we will describe a new proof of the quantitative version of the anisotropic isoperimetric inequality with the sharp exponent; our proof makes use of recent ideas which has led to the sharp isoperimetric inequality in the euclidean case, and of some tools coming from mass transportation.