In probability theory, Bobkov's inequality is a

functional Functional may refer to: * Movements in architecture: ** Functionalism (architecture) ** Form follows function * Functional group, combination of atoms within molecules * Medical conditions without currently visible organic basis: ** Functional sy ...

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 Sergey Bobkov (Russian: Cергей Германович Бобков; born March 15, 1961) is a mathematician. Currently Bobkov is a professor at the University of Minnesota, Twin Cities. He was born in Vorkuta ( Komi Republic, Russia) and grad ...

Bobkov's inequality

Notation: Let *

\gamma^n(dx)=(2\pi)^e^d^nx

be the canonical Gaussian measure on

\R^n

with respect to the

Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...

, *

\phi(x)=(2\pi)^e^

be the one dimensional canonical Gaussian density *

\Phi(t)=\gamma^1 \infty,t /math> the cumulative distribution function
* I(t):=\phi(\Phi^(t)) be a function I(t):,1 to,1 /math> that vanishes at the end points \lim\limits_ I(t)=\lim\limits_ I(t)=0.

Statement

For every locally Lipschitz continuous (or smooth) function

f:\R^n\to,1 /math> the following inequality holds

: I\left( \int_ f d\gamma^n(dx)\right)\leq \int_ \sqrtd\gamma^n(dx).

Generalizations

There exists a generalization by Dominique Bakry and Michel Ledoux.

References

Probabilistic inequalities {{improve categories, date=June 2023