**Aldo Pratelli**

University of Pavia

Pavia, Italy
`pratelli@unipv.it`

**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.