- Gaussian isoperimetric inequality
The Gaussian isoperimetric inequality, proved by
Boris Tsirelson andVladimir Sudakov and independently byChrister Borell , states that among all sets of givenGaussian measure in the "n"-dimensionalEuclidean space ,half-space s have the minimal Gaussian boundary measure.Mathematical formulation
Let be a
measurable subset of endowed with the Gaussian measure γ"n". Denote by :the ε-extension of "A". Then the "Gaussian isoperimetric inequality" states that
:
where
:
Remarks on the proofs
The original proofs by Sudakov, Tsirelson and Borell were based on
Paul Lévy 'sspherical 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 thesemigroup techniques which works in a much more abstract setting. Later Barthe and Maurey gave yet another proof using theBrownian motion .The Gaussian isoperimetric inequality also follows from
Ehrhard's inequality (cf. Latała [6] , Borell [7] ).See also
*
Concentration of measure References
[1] 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[2] Ch. Borell, "The Brunn-Minkowski inequality in Gauss space", Invent. Math. 30 (1975), no. 2, 207--216.
[3] 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
[4] D.Bakry, M.Ledoux, "Lévy-Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator", Invent. Math. 123 (1996), no. 2, 259--281
[5] F. Barthe, B. Maurey, "Some remarks on isoperimetry of Gaussian type", Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 4, 419--434.
[6] R. Latała, "A note on the Ehrhard inequality", Studia Math. 118 (1996), no. 2, 169--174.
[7] Ch. Borell, "The Ehrhard inequality", C. R. Math. Acad. Sci. Paris 337 (2003), no. 10, 663--666.
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