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In this paper, we prove the isoperimetric inequality for the anisotropicGaussian measure and characterize the cases of equality. We also find anexample that shows Ehrhard symmetrization fails to decrease for the anisotropicGaussian perimeter and gives a new inequality that includes an error term. Thisnew inequality, in particular, gives us a hint to prove a uniqueness result forthe anisotropic Ehrhard symmetrization.

The Anisotropic Gaussian Isoperimetric Inequality and Ehrhard Symmetrization