Gaussian isoperimetric inequality

Let $\backslash scriptstyle\; A$ be a measurable subset of $\backslash scriptstyle\backslash mathbf\{R\}^n$ endowed with the standard Gaussian measure $\backslash gamma^n$ with the density $\{\backslash exp(-\backslash |x\backslash |^2/2)\}/(2\backslash pi)^\{n/2\}$. Denote by

: $A\_\backslash varepsilon\; =\; \backslash left\backslash \{\; x\; \backslash in\; \backslash mathbf\{R\}^n\; \backslash ,\; |\; \backslash ,$

\text{dist}(x, A) \leq \varepsilon \right\}

the ε-extension of A. Then the Gaussian isoperimetric inequality states that

: $\backslash liminf\_\{\backslash varepsilon\; \backslash to\; +0\}$

\varepsilon^{-1} \left\{ \gamma^n (A_\varepsilon) - \gamma^n(A) \right\}

\geq \varphi(\Phi^{-1}(\gamma^n(A))),

where

: $\backslash varphi(t)\; =\; \backslash frac\{\backslash exp(-t^2/2)\}\{\backslash sqrt\{2\backslash pi\}\}\backslash quad\{\backslash rm\; and\}\backslash quad\backslash Phi(t)\; =\; \backslash int\_\{-\backslash infty\}^t\; \backslash varphi(s)\backslash ,\; ds.$

The original proofs by Sudakov, Tsirelson and Borell were based on Paul Lévy's spherical isoperimetric inequality.

Sergey Bobkov proved Bobkov's inequality, a functional generalization of the Gaussian isoperimetric inequality, proved from a certain "two point analytic inequality".{{Cite journal|last=Bobkov|first=S. G.|date=1997|title=An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space|journal=The Annals of Probability|language=en|volume=25|issue=1|pages=206–214|doi=10.1214/aop/1024404285|issn=0091-1798|doi-access=free}} Bakry and Ledoux gave another proof of Bobkov's functional inequality based on the semigroup techniques which works in a much more abstract setting.{{Cite journal|last1=Bakry|first1=D.|last2=Ledoux|first2=M.|date=1996-02-01|title=Lévy–Gromov's isoperimetric inequality for an infinite dimensional diffusion generator|journal=Inventiones Mathematicae|language=en|volume=123|issue=2|pages=259–281|doi=10.1007/s002220050026|s2cid=120433074 |issn=1432-1297}} Later Barthe and Maurey gave yet another proof using the Brownian motion.{{Cite journal|last1=Barthe|first1=F.|last2=Maurey|first2=B.|date=2000-07-01|title=Some remarks on isoperimetry of Gaussian type|journal=Annales de l'Institut Henri Poincaré B|volume=36|issue=4|pages=419–434|doi=10.1016/S0246-0203(00)00131-X|bibcode=2000AIHPB..36..419B |issn=0246-0203|url=http://www.numdam.org/item/AIHPB_2000__36_4_419_0/}}

The Gaussian isoperimetric inequality also follows from Ehrhard's inequality.{{Cite journal|last=Latała|first=Rafał|date=1996|title=A note on the Ehrhard inequality|url=https://www.infona.pl//resource/bwmeta1.element.bwnjournal-article-smv118i2p169bwm|journal=Studia Mathematica|language=English|volume=2|issue=118|pages=169–174|doi=10.4064/sm-118-2-169-174 |issn=0039-3223|doi-access=free}}{{Cite journal|last=Borell|first=Christer|date=2003-11-15|title=The Ehrhard inequality|journal=Comptes Rendus Mathématique|volume=337|issue=10|pages=663–666|doi=10.1016/j.crma.2003.09.031|issn=1631-073X}}

• Concentration of measure
• Borell–TIS inequality

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Category:Probabilistic inequalities