Best constants for the isoperimetric inequality in quantitative form

Abstract

We prove some results in the context of isoperimetric inequalities with quantitative terms. In the $2$-dimensional case, our main contribution is a method for determining the optimal coefficients $c_1,\dots ,c_m$ in the inequality $\delta P(E)\ge {\sum }_{k=1}^m{c}_k\alpha (E{)}^k+o(\alpha (E{)}^m)$, valid for each Borel set $E$ with positive and finite area, with $\delta P(E)$ and $\alpha (E)$ being, respectively, the \textit{isoperimetric deficit} and the \textit{Fraenkel asymmetry} of $E$. In $n$ dimensions, besides proving existence and regularity properties of minimizers for a wide class of \textit{quantitative isoperimetric quotients} including the lower semicontinuous extension of $\frac{\delta P(E)}{\alpha (E{)}^2}$, we describe the general technique upon which our $2$-dimensional result is based. This technique, called Iterative Selection Principle, extends the one introduced in [12].