  In this note, we provide an adaptation of the Kohler-Jobin rearrangement
technique to the setting of the Gauss space. As a result, we prove the Gaussian
analogue of the Kohler-Jobin's resolution of a conjecture of P\'{o}lya-Szeg\"o:
when the Gaussian torsional rigidity of a (convex) domain is fixed, the
Gaussian principal frequency is minimized for the half-space. At the core of
this rearrangement technique is the idea of considering a "modified" torsional
rigidity, with respect to a given function, and rearranging its layers to
half-spaces, in a particular way; the Rayleigh quotient decreases with this
procedure.
  We emphasize that the analogy of the Gaussian case with the Lebesgue case is
not to be expected here, as in addition to some soft symmetrization ideas, the
argument relies on the properties of some special functions; the fact that this
analogy does hold is somewhat of a miracle.
