  We study an isoperimetric problem described by a functional that consists of
the standard Gaussian perimeter and the norm of the barycenter. This second
term has a repulsive effect, and it is in competition with the perimeter.
Because of that, in general the solution is not the half-space. We characterize
all the minimizers of this functional, when the volume is close to one, by
proving that the minimizer is either the half-space or the symmetric strip,
depending on the strength of the repulsive term. As a corollary, we obtain that
the symmetric strip is the solution of the Gaussian isoperimetric problem among
symmetric sets when the volume is close to one.
