|
|
Line 1: |
Line 1: |
| 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]].
| | 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.<br><br>My blog post: [http://forsaken-ranger.5cz.de/index.php?mod=users&action=view&id=1945 extended warranty auto] |
| | |
| == Mathematical formulation ==
| |
| 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
| |
| : <math>A_\varepsilon = \left\{ x \in \mathbf{R}^n \, | \,
| |
| \text{dist}(x, A) \leq \varepsilon \right\}</math>
| |
| | |
| the ε-extension of ''A''. Then the ''Gaussian isoperimetric inequality'' states that
| |
| | |
| : <math>\liminf_{\varepsilon \to +0}
| |
| \varepsilon^{-1} \left\{ \gamma^n (A_\varepsilon) - \gamma^n(A) \right\}
| |
| \geq \varphi(\Phi^{-1}(\gamma^n(A))),</math>
| |
| | |
| where
| |
| | |
| : <math>\varphi(t) = \frac{\exp(-t^2/2)}{\sqrt{2\pi}}\quad{\rm and}\quad\Phi(t) = \int_{-\infty}^t \varphi(s)\, ds. </math>
| |
| | |
| == Remarks on the proofs ==
| |
| 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]].
| |
| | |
| The Gaussian isoperimetric inequality also follows from [[Ehrhard's inequality]] (cf. Latała, Borell).
| |
| | |
| == See also == | |
| * [[Concentration of measure]]
| |
| | |
| ==References==
| |
| * 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
| |
| * Ch. Borell, ''The Brunn-Minkowski inequality in Gauss space'', Invent. Math. 30 (1975), no. 2, 207–216.
| |
| * 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
| |
| * D.Bakry, M.Ledoux, ''Lévy-Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator'', Invent. Math. 123 (1996), no. 2, 259–281
| |
| * F. Barthe, B. Maurey, ''Some remarks on isoperimetry of Gaussian type'', Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 4, 419–434.
| |
| * R. Latała, ''A note on the Ehrhard inequality'', Studia Math. 118 (1996), no. 2, 169–174.
| |
| * Ch. Borell, ''The Ehrhard inequality'', C. R. Math. Acad. Sci. Paris 337 (2003), no. 10, 663–666.
| |
| | |
| [[Category:Probabilistic inequalities]]
| |
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.
My blog post: extended warranty auto