# Sharp quantitative stability for isoperimtric inequalities with homogeneous weights

created by pratelli on 29 Jun 2020
modified on 02 Sep 2020

[BibTeX]

Preprint

Inserted: 29 jun 2020
Last Updated: 2 sep 2020

Year: 2020

ArXiv: 2006.13867 PDF

Abstract:

We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights.

Inspired by the proof of such isoperimetric inequalities through the ABP method (see \cite{CRS}), we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set $E$ and the minimizer of the inequality (as in Gromov's proof of the isoperimetric inequality).

Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of \cite{FMP} and prove that if $E$ is almost optimal for the inequality then it is quantitatively close to a minimizer \emph{up to translations}. Then, a delicate analysis is necessary to rule out the possibility of translations.

As a step of our proof, we establish a sharp regularity result for \emph{restricted} convex envelopes of a function that might be of independent interest.

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