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 (or smooth) function the following inequality holds

Generalizations

There exists a generalization by Dominique Bakry and Michel Ledoux.

References