A Selection Principle for the Sharp Quantitative Isoperimetric Inequality

Abstract. We introduce a new variational method for the study of stability in the isoperimetric inequal-ity. The method is quite general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter. Two applications are presented. First we give a new proof of the sharp quantitative isoperimetric inequality in Rn. Second we positively answer to a conjecture by Hall concerning the best constant for the quantitative isoperimetric inequality in R2 in the small asymmetry regime. 1