  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. On the other hand, in
dimension two and higher, the ball or its complement does not always maximize
noise stability among symmetric sets of fixed Gaussian measure. In summary, we
provide the first known positive and negative results for the Symmetric
Gaussian Problem.
