A selection principle for the sharp quantitative isoperimetric inequality

We introduce a new variational method for the study of isoperimetric inequalities with quantitative terms.The method is general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter. Two notable applications are presented. First we give a new proof of the sharp quantitative isoperimetric inequality in R^n. Second we positively answer to a conjecture by Hall concerning the best constant for the quantitative isoperimetric inequality in R^2 in the small asymmetry regime.