  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.
