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 Russia mathematician Sergey Bobkov.

Bobkov's inequality

Notation:

Let

• [math]\displaystyle{ \gamma^n(dx)=(2\pi)^{-n/2}e^{-\|x\|^2/2}d^nx }[/math] be the canonical Gaussian measure on [math]\displaystyle{ \R^n }[/math] with respect to the Lebesgue measure,
• [math]\displaystyle{ \phi(x)=(2\pi)^{-1/2}e^{-x^2/2} }[/math] be the one dimensional canonical Gaussian density
• [math]\displaystyle{ \Phi(t)=\gamma^1[-\infty,t] }[/math] the cumulative distribution function
• [math]\displaystyle{ I(t):=\phi(\Phi^{-1}(t)) }[/math] be a function [math]\displaystyle{ I(t):[0,1]\to [0,1] }[/math] that vanishes at the end points [math]\displaystyle{ \lim\limits_{t\to 0} I(t)=\lim\limits_{t\to 1} I(t)=0. }[/math]

Statement

For every locally Lipschitz continuous (or smooth) function [math]\displaystyle{ f:\R^n\to[0,1] }[/math] the following inequality holds

[math]\displaystyle{ I\left( \int_{\R^n} f d\gamma^n(dx)\right)\leq \int_{\R^n} \sqrt{I(f)^2+|\nabla f|^2}d\gamma^n(dx). }[/math]

Generalizations

There exist a generalization by Dominique Bakry und Michel Ledoux.