# Best constants for the isoperimetric inequality in quantitative form

### Marco Cicalese

Università degli Studi di Napoli Federico II, Italy### Gian Paolo Leonardi

Università di Modena e Reggio Emilia, Italy

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

## Cite this article

Marco Cicalese, Gian Paolo Leonardi, Best constants for the isoperimetric inequality in quantitative form. J. Eur. Math. Soc. 15 (2013), no. 3, pp. 1101–1129

DOI 10.4171/JEMS/387