Sharp quantitative stability for isoperimetric inequalities with homogeneous weights

TL;DR

This paper establishes the sharp quantitative stability of weighted isoperimetric inequalities in convex cones with homogeneous weights, using a novel convex coupling and regularity results to show near-optimal sets are close to minimizers.

Contribution

It introduces a new convex coupling method for stability analysis in weighted isoperimetric inequalities, extending previous techniques to include weights and convex cones.

Findings

Proves sharp quantitative stability for weighted isoperimetric inequalities.

Develops a new convex coupling approach inspired by the ABP method.

Establishes a sharp regularity result for convex envelopes.

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, 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 Figalli-Maggi-Pratelli and prove that if $E$ is almost optimal for the inequality then it is quantitatively close to a minimizer 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 restricted convex envelopes of a function that might be of independent interest.