Low Correlation Noise Stability of Symmetric Sets

We study the Gaussian noise stability of subsets A of Euclidean space satisfying A = −A. It is shown that an interval centered at the origin, or its complement, maximizes noise stability for small correlation, among symmetric subsets of the real line of fixed Gaussian measure. In the plane, we show that a ball or its complement locally maximizes noise stability for small correlation, among symmetric sets of fixed Gaussian volume. Some asymptotic results are proven in higher-dimensional Euclidean space. In summary, we provide the first known positive results for the Symmetric Gaussian Problem.