Title: A Selection Principle for the Sharp Quantitative Isoperimetric Inequality
Abstract: 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 $${\mathbb{R}^n}$$ . Second we positively answer a conjecture by Hall concerning the best constant for the quantitative isoperimetric inequality in $${\mathbb{R}^2}$$ in the small asymmetry regime.