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

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

Statement

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

$I(\underset{{ℝ}^n}{\int }fd{\gamma }^n(dx))\le \underset{{ℝ}^n}{\int }\sqrt{I(f{)}^2+|\mathrm{\nabla }f{|}^2}d{\gamma }^n(dx).$

Generalizations

There exists a generalization by Dominique Bakry and Michel Ledoux.