Bobkov's inequality

In probability theory, Bobkov's inequality is a functional isoperimetric inequality for the canonical Gaussian measure. It generalizes the Gaussian isoperimetric inequality.
The equation was proven in 1997 by the Russian mathematician Sergey Bobkov.

Bobkov's inequality

Notation:
Let

be the canonical Gaussian measure on with respect to the Lebesgue measure,
be the one dimensional canonical Gaussian density
the cumulative distribution function
be a function that vanishes at the end points

Statement

For every locally Lipschitz continuous function the following inequality holds

Generalizations

There exists a generalization by Dominique Bakry and Michel Ledoux.