Calculus of Variations and Geometric Measure Theory

M. Cicalese - G. P. Leonardi

A Selection Principle for the Sharp Quantitative Isoperimetric Inequality

created by leonardi on 22 Jul 2010
modified on 31 Jan 2024


Published Paper

Inserted: 22 jul 2010
Last Updated: 31 jan 2024

Journal: Arch. Rat. Mech. Anal.
Volume: 206
Number: 2
Pages: 617-643
Year: 2012

ArXiv: 1007.3899 PDF


We introduce a new variational method for the study of stability in the isoperimetric inequality. The method is quite general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter. Two applications are presented. First we give a new proof of the sharp quantitative isoperimetric inequality in $R^n$. Second we positively answer to a conjecture by Hall concerning the best constant for the quantitative isoperimetric inequality in $R^2$ in the small asymmetry regime.