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October 12, 2016 thcscience_admin

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[edit]

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[edit]

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[edit]

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

References[edit]

^ Bobkov, Sergey G. (1997). "An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space". The Annals of Probability. 25 (1). Institute of Mathematical Statistics: 206–214. doi:10.1214/aop/1024404285. S2CID 120975922.
^ Bobkov, Sergey G. (1997). "An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space". The Annals of Probability. 25 (1). Institute of Mathematical Statistics: 209. doi:10.1214/aop/1024404285. S2CID 120975922.
^ Carlen, Eric; Kerce, James (2001). "On the case of equality in Bobkov's inequality and Gaussian rearrangement". Calculus of Variations. 13: 2. doi:10.1007/PL00009921. S2CID 119968388.
^ Bakry, Dominique; Ledoux, Michel (1996). "Lévy–Gromov's isoperimetric inequality for an infinite dimensional diffusion generator". Inventiones Mathematicae. 123 (2): 259–281. doi:10.1007/s002220050026. S2CID 120433074.

source: https://en.wikipedia.org/wiki/Bobkov%27s_inequality

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