A two-sided estimate for the Gaussian noise stability deficit

The Gaussian noise-stability of a set $$A \subset {\mathbb R}^n$$A⊂Rn is defined by $$ \begin{aligned} {\mathcal {S}}_\rho (A) = {\mathbb P}\left( X \in A ~ \& ~ Y \in A \right) \end{aligned}$$Sρ(A)=PX∈A&Y∈Awhere $$X,Y$$X,Y are standard jointly Gaussian vectors satisfying $${\mathbb E}[X_i Y_j] = \delta _{ij} \rho $$E[XiYj]=δijρ. Borell’s inequality states that for all $$0 < \rho < 1$$0<ρ<1, among all sets $$A \subset {\mathbb R}^n$$A⊂Rn with a given Gaussian measure, the quantity $${\mathcal {S}}_\rho (A)$$Sρ(A) is maximized when $$A$$A is a half-space. We give a novel short proof of this fact, based on stochastic calculus. Moreover, we prove an almost tight, two-sided, dimension-free robustness estimate for this inequality: by introducing a new metric to measure the distance between the set $$A$$A and its corresponding half-space $$H$$H (namely the distance between the two centroids), we show that the deficit $${\mathcal {S}}_\rho (H) - {\mathcal {S}}_\rho (A)$$Sρ(H)-Sρ(A) can be controlled from both below and above by essentially the same function of the distance, up to logarithmic factors. As a consequence, we also establish the conjectured exponent in the robustness estimate proven by Mossel-Neeman, which uses the total-variation distance as a metric. In the limit $$\rho \rightarrow 1$$ρ→1, we obtain an improved dimension-free robustness bound for the Gaussian isoperimetric inequality. Our estimates are also valid for a generalized version of stability where more than two correlated vectors are considered.