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| In mathematics, the '''Gaussian isoperimetric inequality''', proved by [[Boris Tsirelson]] and [[Vladimir Sudakov]] and independently by [[Christer Borell]], states that among all sets of given [[Gaussian measure]] in the ''n''-dimensional [[Euclidean space]], [[Half-space (geometry)|half-space]]s have the minimal Gaussian [[Minkowski content|boundary measure]].
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| == Mathematical formulation ==
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| Let <math>\scriptstyle A</math> be a [[measurable]] subset of <math>\scriptstyle\mathbf{R}^n </math> endowed with the Gaussian measure γ<sup> ''n''</sup>. Denote by
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| : <math>A_\varepsilon = \left\{ x \in \mathbf{R}^n \, | \,
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| \text{dist}(x, A) \leq \varepsilon \right\}</math>
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| the ε-extension of ''A''. Then the ''Gaussian isoperimetric inequality'' states that
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| : <math>\liminf_{\varepsilon \to +0}
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| \varepsilon^{-1} \left\{ \gamma^n (A_\varepsilon) - \gamma^n(A) \right\}
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| \geq \varphi(\Phi^{-1}(\gamma^n(A))),</math>
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| where
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| : <math>\varphi(t) = \frac{\exp(-t^2/2)}{\sqrt{2\pi}}\quad{\rm and}\quad\Phi(t) = \int_{-\infty}^t \varphi(s)\, ds. </math>
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| == Remarks on the proofs ==
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| The original proofs by Sudakov, Tsirelson and Borell were based on [[Paul Lévy (mathematician)|Paul Lévy]]'s [[spherical isoperimetric inequality]]. Another approach is due to Bobkov, who introduced a functional inequality generalizing the Gaussian isoperimetric inequality and derived it from a certain two-point inequality. Bakry and Ledoux gave another proof of Bobkov's functional inequality based on the [[semigroup]] techniques which works in a much more abstract setting. Later Barthe and Maurey gave yet another proof using the [[Brownian motion]].
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| The Gaussian isoperimetric inequality also follows from [[Ehrhard's inequality]] (cf. Latała, Borell).
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| == See also == | |
| * [[Concentration of measure]]
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| ==References==
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| * V.N.Sudakov, B.S.Cirelson [Tsirelson], ''Extremal properties of half-spaces for spherically invariant measures'', (Russian) Problems in the theory of probability distributions, II, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. ([[LOMI]]) 41 (1974), 14–24, 165
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| * Ch. Borell, ''The Brunn-Minkowski inequality in Gauss space'', Invent. Math. 30 (1975), no. 2, 207–216.
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| * S.G.Bobkov, ''An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space'', Ann. Probab. 25 (1997), no. 1, 206–214
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| * D.Bakry, M.Ledoux, ''Lévy-Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator'', Invent. Math. 123 (1996), no. 2, 259–281
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| * F. Barthe, B. Maurey, ''Some remarks on isoperimetry of Gaussian type'', Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 4, 419–434.
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| * R. Latała, ''A note on the Ehrhard inequality'', Studia Math. 118 (1996), no. 2, 169–174.
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| * Ch. Borell, ''The Ehrhard inequality'', C. R. Math. Acad. Sci. Paris 337 (2003), no. 10, 663–666.
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| [[Category:Probabilistic inequalities]]
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Nestor is the title my parents gave me but I don't like when people use my full name. Some time ago he selected to reside in Kansas. My occupation is a manufacturing and distribution officer and I'm doing pretty great financially. One of the things I adore most is greeting card gathering but I don't have the time recently.
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