Bobkov's inequality

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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.^[1]

Bobkov's inequality

Notation:

Let

$\gamma ^{n}(dx)=(2\pi )^{-n/2}e^{-\|x\|^{2}/2}d^{n}x$ be the canonical Gaussian measure on $\mathbb {R} ^{n}$ 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 $f:\mathbb {R} ^{n}\to [0,1]$ the following inequality holds^[2]^[3]

I\left(\int _{\mathbb {R} ^{n}}fd\gamma ^{n}(dx)\right)\leq \int _{\mathbb {R} ^{n}}{\sqrt {I(f)^{2}+|\nabla f|^{2}}}d\gamma ^{n}(dx).

Generalizations

There exists a generalization by Dominique Bakry and Michel Ledoux.^[4]

References

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