  In this paper we study the functional given by the integral of the mean
curvature of a convex set with Gaussian weight with Gaussian volume constraint.
It was conjectured that the ball centered at the origin is the only minimizer
of such a functional for certain value of the mass. We give a positive answer
in dimension two while in higher dimension the situation is different. In fact,
for small value of mass the ball centered at the origin is a local minimizer
while for large values the ball is a maximizer among convex sets with uniform
bound on the curvature.
