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\[\\delta P(E) \geq \sum_{k=1}^{m}c_{k}\alpha(E)^{k} + o(\alpha(E)^{m}),\]valid for each Borel set $E$ with positive and finite area, with $\delta(E)$ and $\alpha(E)$ being, respectively, the \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 \cite{CicLeo10}.