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In this note, we provide an adaptation of the Kohler-Jobin rearrangementtechnique to the setting of the Gauss space. As a result, we prove the Gaussiananalogue 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, theGaussian principal frequency is minimized for the half-space. At the core ofthis rearrangement technique is the idea of considering a "modified" torsionalrigidity, with respect to a given function, and rearranging its layers tohalf-spaces, in a particular way; the Rayleigh quotient decreases with thisprocedure.  We emphasize that the analogy of the Gaussian case with the Lebesgue case isnot to be expected here, as in addition to some soft symmetrization ideas, theargument relies on the properties of some special functions; the fact that thisanalogy does hold is somewhat of a miracle.

Kohler-Jobin meets Ehrhard: the sharp lower bound for the Gaussian  principal frequency while the Gaussian torsional rigidity is fixed, via  rearrangements