Best constants for the isoperimetric inequality in quantitative form

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 $c_1,...,c_m$ in the inequality\[\\delta P(E) \geq \sum_k=1^mc_k\alpha(E)^k + o(\alpha(E)^m),\]valid for each Borel set $E$ with positive and finite area, with $\delta(E)$ and $\alpha(E)$ being, respectively, the \textitisoperimetric deficit and the \textitFraenkel asymmetry of $E$. In $n$ dimensions, besides proving existence and regularity properties of minimizers for a wide class of \textitquantitative 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 \citeCicLeo10.