A Functional Form of the Isoperimetric Inequality for the Gaussian Measure

Letgbe a smooth function on Rnwith values in [0, 1]. Using the isoperimetric property of the Gaussian measure, itis proved thatφ(Φ−1(Eg))−Eφ(Φ−1(g))⩽E|∇g|. Conversely, this inequality implies the isoperimetric property of the Gaussian measure.