A Selection Principle for the Sharp Quantitative Isoperimetric Inequality

By Marco Cicalese, Gian Paolo Leonardi. Archive for Rational Mechanics and Analysis, 2010

216 citations recorded by the source corpus.

We introduce a new variational method for the study of isoperimetric inequalities with quantitative terms. The method is general as it relies on a penalization technique combined with the regularity theory for quasiminimizers of the perimeter. Two notable applications are presented. First we give a new proof of the sharp quantitative isoperimetric inequality in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^n}$$\end{document}. Second we positively answer a conjecture by Hall concerning the best constant for the quantitative isoperimetric inequality in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}^2}$$\end{document} in the small asymmetry regime.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/013d929b61c4d5d50dd965b7e2786e3da6a05901

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In Applied Mathematics

Citations in the corpus

A strong form of the quantitative isoperimetric inequality (2011, methodology): “In order to prove this inequality we follow the strategy introduced in [3] for proving the quantitative inequality stated in (1.”
The quantitative isoperimetric inequality and related topics (2015, methodology): “The value of the constant in dimension n = 2 for small values of the asymmetry was first given by Hall, Hayman and Weitsman in [84, 83] in the convex case, and then extended by Cicalese and Leonardi in [45] for general sets of finite perimeter.”
Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence (2022, background): “In fact, stability has attracted a lot of attention in recent years and we refer to [44, 26, 39, 27, 23, 30, 21, 24, 38, 25, 61, 42, 43, 40, 10] and the references within for a list of works in this direction.”
Sharp and rigid isoperimetric inequalities in metric-measure spaces with lower Ricci curvature bounds (2015, background): “The isoperimetric problem has been analyzed from several complementary points of view: for an overview of the more geometric aspects we refer to [55, 61, 62], for the approach via geometric measure theory see for instance [47, 51], for the connections with convex and integral geometry see [12], for ”
Sharp gradient stability for the Sobolev inequality (2020, background): “Introduction Motivated by important applications to problems in the calculus of variations and evolution PDEs, in recent years there has been a growing interest around the understanding of quantitative stability for functional/geometric inequalities, see for instance [3, 2, 8, 27, 28, 21, 9, 22, 29,”
Stability of the Steiner symmetrization of convex sets (2013, background): “In recent years quantitative isoperimetric and related functional inequalities have attracted the interest of several authors ([8, 11, 12, 13, 14, 15], to name but a few).”
Droplet Minimizers of an Isoperimetric Problem with Long‐Range Interactions (2011, background): “The following, called sharp quantitative isoperimetric inequality, has been proved in [17, 23, 25].”
Faber–Krahn inequalities in sharp quantitative form (2013, methodology): “The selection principle turns out to be a very flexible technique and after the paper [15] it has been applied to a wide variety of geometric problems, see for instance [1, 8] and [17].”
Isoperimetry and Stability Properties of Balls with Respect to Nonlocal Energies (2014, methodology): “[11] Cicalese, M.; Leonardi, G. P.; A Selection Principle for the Sharp Quantitative Isoperimetric Inequality.”