LA Probability Forum
The symmetric Gaussian isoperimetric problem of Barthe (2001) asks for the symmetric Euclidean set of fixed Gaussian volume with smallest Gaussian surface area. We show that, if such a set is a cylinder, then this set or its complement must be convex. Moreover, except for one case for the sign of the Gaussian mean curvature of the optimal set in R^n, the boundary of the optimal set must be r S^k × R^(n-k) for some (n-1)^(1/2) ≤ r ≤ (n+1)^(1/2) and for some 1 ≤ k ≤ n-1 (if k=0 we do not constrain r). It has been known since the 1970s that a Euclidean set with fixed Gaussian volume and smallest Gaussian surface area is a half space, and this implies e.g. Gaussian concentration of measure. The added symmetry assumption of Barthe's problem introduces extra difficulties.