Robust Optimality of Gaussian Noise Stability

TL;DR

This paper proves that under Gaussian measure, half-spaces are uniquely the most noise stable sets, and provides a quantitative stability result showing near-optimal sets are close to half-spaces, extending Borell's theorem and answering Ledoux's question.

Contribution

It establishes the uniqueness of half-spaces as the most noise stable sets under Gaussian measure and provides a quantitative stability version of this result.

Findings

Half-spaces are uniquely the most noise stable under Gaussian measure.

Near-optimal sets are quantitatively close to half-spaces.

Extends Borell's theorem with a quantitative stability version.

Abstract

We prove that under the Gaussian measure, half-spaces are uniquely the most noise stable sets. We also prove a quantitative version of uniqueness, showing that a set which is almost optimally noise stable must be close to a half-space. This extends a theorem of Borell, who proved the same result but without uniqueness, and it also answers a question of Ledoux, who asked whether it was possible to prove Borell's theorem using a direct semigroup argument. Our quantitative uniqueness result has various applications in diverse fields.