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.
Cite this article
Elchanan Mossel, Joe Neeman, Robust optimality of Gaussian noise stability. J. Eur. Math. Soc. 17 (2015), no. 2, pp. 433–482DOI 10.4171/JEMS/507