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

A_{\varepsilon }=\left\{x\in {\mathbf {R}}^{n}\,|\,{\text{dist}}(x,A)\leq \varepsilon \right\}

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

\liminf _{{\varepsilon \to +0}}\varepsilon ^{{-1}}\left\{\gamma ^{n}(A_{\varepsilon })-\gamma ^{n}(A)\right\}\geq \varphi (\Phi ^{{-1}}(\gamma ^{n}(A))),

where

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

Proofs and generalizations

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

Sergey Bobkov proved a functional generalization of the Gaussian isoperimetric inequality, 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]

References

^ Sudakov, V. N.; Tsirel'son, B. S. (1978-01-01) [Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 41, pp. 14–24, 1974]. "Extremal properties of half-spaces for spherically invariant measures". 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.
^ Bobkov, S. G. (1997). "An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space". 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". 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.
^ Latała, Rafał (1996). "A note on the Ehrhard inequality". Studia Mathematica. 2 (118): 169–174. ISSN 0039-3223.
^ 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.

[1] Sudakov, V. N.; Tsirel'son, B. S. (1978-01-01) [Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 41, pp. 14–24, 1974]. "Extremal properties of half-spaces for spherically invariant measures". Journal of Soviet Mathematics. 9 (1): 9–18. doi:10.1007/BF01086099. ISSN 1573-8795.

[2] Borell, Christer (1975). "The Brunn-Minkowski Inequality in Gauss Space". Inventiones Mathematicae. 30 (2): 207–216. doi:10.1007/BF01425510. ISSN 0020-9910.

[3] Bobkov, S. G. (1997). "An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space". The Annals of Probability. 25 (1): 206–214. doi:10.1214/aop/1024404285. ISSN 0091-1798.

[4] Bakry, D.; Ledoux, M. (1996-02-01). "Lévy–Gromov's isoperimetric inequality for an infinite dimensional diffusion generator". Inventiones Mathematicae. 123 (2): 259–281. doi:10.1007/s002220050026. ISSN 1432-1297.

[5] 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.

[6] Latała, Rafał (1996). "A note on the Ehrhard inequality". Studia Mathematica. 2 (118): 169–174. ISSN 0039-3223.

[7] 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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Gaussian isoperimetric inequality

Mathematical formulation

Proofs and generalizations

See also

References