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 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

• 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. S2CID 121935322.
2. Borell, Christer (1975). "The Brunn-Minkowski Inequality in Gauss Space". Inventiones Mathematicae. 30 (2): 207–216. Bibcode:1975InMat..30..207B. doi:10.1007/BF01425510. ISSN 0020-9910. S2CID 119453532.
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. S2CID 120433074.
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. Bibcode:2000AIHPB..36..419B. 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. doi:10.4064/sm-118-2-169-174. 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.