The Gaussian noise-stability of a set A⊂Rn\documentclass[12pt]{minimal}
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\begin{document}$$A \subset {\mathbb R}^n$$\end{document} is defined by Sρ(A)=PX∈A&Y∈A\documentclass[12pt]{minimal}
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\begin{document}$$ \begin{aligned} {\mathcal {S}}_\rho (A) = {\mathbb P}\left( X \in A ~ \& ~ Y \in A \right) \end{aligned}$$\end{document}where X,Y\documentclass[12pt]{minimal}
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\begin{document}$$X,Y$$\end{document} are standard jointly Gaussian vectors satisfying E[XiYj]=δijρ\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb E}[X_i Y_j] = \delta _{ij} \rho $$\end{document}. Borell’s inequality states that for all 0<ρ<1\documentclass[12pt]{minimal}
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\begin{document}$$0 < \rho < 1$$\end{document}, among all sets A⊂Rn\documentclass[12pt]{minimal}
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\begin{document}$$A \subset {\mathbb R}^n$$\end{document} with a given Gaussian measure, the quantity Sρ(A)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {S}}_\rho (A)$$\end{document} is maximized when A\documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document} and its corresponding half-space H\documentclass[12pt]{minimal}
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\begin{document}$$H$$\end{document} (namely the distance between the two centroids), we show that the deficit Sρ(H)-Sρ(A)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {S}}_\rho (H) - {\mathcal {S}}_\rho (A)$$\end{document} 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 ρ→1\documentclass[12pt]{minimal}
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\begin{document}$$\rho \rightarrow 1$$\end{document}, 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.
