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

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

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

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

where

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

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].

See also

• Concentration of measure
• Borell–TIS inequality

References

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 Mathematique. 337 (10): 663–666. doi:10.1016/j.crma.2003.09.031. ISSN 1631-073X.