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

Statement
For every locally Lipschitz continuous (or smooth) function $$f:\R^n\to[0,1]$$ the following inequality holds


 * $$I\left( \int_{\R^n} f d\gamma^n(dx)\right)\leq \int_{\R^n} \sqrt{I(f)^2+|\nabla f|^2}d\gamma^n(dx).$$

Generalizations
There exists a generalization by Dominique Bakry and Michel Ledoux.