# Best constants for the isoperimetric inequality in quantitative form

Marco Cicalese; Gian Paolo Leonardi

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 3, page 1101-1129
- ISSN: 1435-9855

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topCicalese, Marco, and Leonardi, Gian Paolo. "Best constants for the isoperimetric inequality in quantitative form." Journal of the European Mathematical Society 015.3 (2013): 1101-1129. <http://eudml.org/doc/277535>.

@article{Cicalese2013,

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 $c_1,\ldots ,c_m$ in the inequality $\delta P(E)\ge \sum ^m_\{k=1\}c_k\alpha (E)^k+o(\alpha (E)^m)$, 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].},

author = {Cicalese, Marco, Leonardi, Gian Paolo},

journal = {Journal of the European Mathematical Society},

keywords = {best constants; isoperimetric inequality; quasiminimizers of the perimeter; best constants; isoperimetric inequality; quasiminimizers of the perimeter},

language = {eng},

number = {3},

pages = {1101-1129},

publisher = {European Mathematical Society Publishing House},

title = {Best constants for the isoperimetric inequality in quantitative form},

url = {http://eudml.org/doc/277535},

volume = {015},

year = {2013},

}

TY - JOUR

AU - Cicalese, Marco

AU - Leonardi, Gian Paolo

TI - Best constants for the isoperimetric inequality in quantitative form

JO - Journal of the European Mathematical Society

PY - 2013

PB - European Mathematical Society Publishing House

VL - 015

IS - 3

SP - 1101

EP - 1129

AB - 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 $c_1,\ldots ,c_m$ in the inequality $\delta P(E)\ge \sum ^m_{k=1}c_k\alpha (E)^k+o(\alpha (E)^m)$, 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].

LA - eng

KW - best constants; isoperimetric inequality; quasiminimizers of the perimeter; best constants; isoperimetric inequality; quasiminimizers of the perimeter

UR - http://eudml.org/doc/277535

ER -

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