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}.

Best constants for the isoperimetric inequality in quantitative form / Cicalese, Marco; Leonardi, Gian Paolo. - In: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY. - ISSN 1435-9855. - STAMPA. - 15:(2013), pp. 1101-1129. [10.4171/JEMS/387]

### 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 $c_{1},...,c_{m}$ 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}.
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Best constants for the isoperimetric inequality in quantitative form / Cicalese, Marco; Leonardi, Gian Paolo. - In: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY. - ISSN 1435-9855. - STAMPA. - 15:(2013), pp. 1101-1129. [10.4171/JEMS/387]
Cicalese, Marco; Leonardi, Gian Paolo
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11380/928890
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