Gaussian isoperimetric inequality
In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov,[1] and later independently by Christer Borell,[2] states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian boundary measure.
Mathematical formulation
Let [math]\displaystyle{ \scriptstyle A }[/math] be a measurable subset of [math]\displaystyle{ \scriptstyle\mathbf{R}^n }[/math] endowed with the standard Gaussian measure [math]\displaystyle{ \gamma^n }[/math] with the density [math]\displaystyle{ {\exp(-\|x\|^2/2)}/(2\pi)^{n/2} }[/math]. Denote by
- [math]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ \varphi(t) = \frac{\exp(-t^2/2)}{\sqrt{2\pi}}\quad{\rm and}\quad\Phi(t) = \int_{-\infty}^t \varphi(s)\, ds. }[/math]
Proofs and generalizations
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".[3] Bakry and Ledoux gave another proof of Bobkov's functional inequality based on the semigroup techniques which works in a much more abstract setting.[4] Later Barthe and Maurey gave yet another proof using the Brownian motion.[5]
The Gaussian isoperimetric inequality also follows from Ehrhard's inequality.[6][7]
See also
References
- ↑ Sudakov, V. N.; Tsirel'son, B. S. (1978-01-01). "Extremal properties of half-spaces for spherically invariant measures" (in en). Journal of Soviet Mathematics 9 (1): 9–18. doi:10.1007/BF01086099. ISSN 1573-8795.
- ↑ Borell, Christer (1975). "The Brunn-Minkowski Inequality in Gauss Space.". Inventiones Mathematicae 30 (2): 207–216. doi:10.1007/BF01425510. ISSN 0020-9910. Bibcode: 1975InMat..30..207B. https://eudml.org/doc/142349.
- ↑ Bobkov, S. G. (1997). "An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space" (in en). The Annals of Probability 25 (1): 206–214. doi:10.1214/aop/1024404285. ISSN 0091-1798.
- ↑ Bakry, D.; Ledoux, M. (1996-02-01). "Lévy–Gromov's isoperimetric inequality for an infinite dimensional diffusion generator" (in en). Inventiones Mathematicae 123 (2): 259–281. doi:10.1007/s002220050026. ISSN 1432-1297.
- ↑ Barthe, F.; Maurey, B. (2000-07-01). "Some remarks on isoperimetry of Gaussian type". Annales de l'Institut Henri Poincaré B 36 (4): 419–434. doi:10.1016/S0246-0203(00)00131-X. ISSN 0246-0203. Bibcode: 2000AIHPB..36..419B. http://www.numdam.org/item/AIHPB_2000__36_4_419_0/.
- ↑ Latała, Rafał (1996). "A note on the Ehrhard inequality" (in English). Studia Mathematica 2 (118): 169–174. doi:10.4064/sm-118-2-169-174. ISSN 0039-3223. https://www.infona.pl//resource/bwmeta1.element.bwnjournal-article-smv118i2p169bwm.
- ↑ Borell, Christer (2003-11-15). "The Ehrhard inequality". Comptes Rendus Mathématique 337 (10): 663–666. doi:10.1016/j.crma.2003.09.031. ISSN 1631-073X.
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