UN NUOVO APPROCCIO ALLE DISUGUAGLIANZE ISOPERIMETRICHE QUANTITATIVE

We introduce a new variational method for studying geometric and functional inequalities with quantitative terms. In the context of isoperimetric-type inequalities, this method (called Selection Principle) is based on a penalization technique combined with the regularity theory of quasiminimizers of the perimeter functional. In this seminar we present the method and describe two remarkable applications. The rst one is a new proof of the sharp quantitative isoperimetric inequality in Rn. The second one is the proof of a conjecture posed by Hall about the optimal constant in the quantitative isoperimetric inequality in R2, in the small asymmetry regime.