Best constants for the isoperimetric inequality in quantitative form

Cicalese, 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 -