The isoperimetric inequality in quantitative form

The classical isoperimetric inequality states that, given a set $E$ in $R^n$ having the same measure of the unit ball $B$, the perimeter $P(E)$ of $E$ is greater than the perimeter $P(B)$ of $B$. Moreover, when the isoperimetric deficit $D(E)=P(E)-P(B)$ equals 0, than $E$ coincides (up to a translation) with $B$. The sharp quantitative form of the isoperimetric inequality states that $D(E)$ can be bound from below by $A(E)^2$, where the Fraenkel asymmetry $A(E)$ of $E$ is defined as the minimum of the volume of the symmetric difference between $E$ and any translation of $B$. This result, conjectured by Hall in 1990, has been proven in its full generality by Fusco-Maggi-Pratelli (Ann. of Math. 2008) via symmetrization arguments and more recently by Figalli-Maggi-Pratelli (Invent. Math. 2010) through optimal transportation techniques. In this talk I will present a new proof of the sharp quantitative version of the isoperimetric inequality that I have recently obtained in collaboration with G.P.Leonardi (University of Modena e Reggio). The proof relies on a variational method in which a penalization technique is combined with the regularity theory for quasiminimizers of the perimeter. As a further application of this method I will present a positive answer to another conjecture posed by Hall in 1992 concerning the best constant for the quantitative isoperimetric inequality in $R^2$ in the small asymmetry regime.