Gaussian isoperimetric inequality

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The Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov and independently by Christer Borell, states that among all sets A \subset \mathbf{R}^n of given Gaussian measure, halfspaces have minimal Gaussian boundary measure.

Equivalently,

\liminf_{\epsilon \to +0} \epsilon^{-1} \left\{ \gamma^n (A_\epsilon) - \gamma^n(A) \right\} \geq \phi(\Phi^{-1}(\gamma^n(A))),

where γn is a Gaussian measure on \mathbf{R}^n,

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

is the ε-extension of A,

\phi(t) = \frac{\exp(-t^2/2)}{\sqrt{2\pi}}

and \Phi(t) = \int_{-\infty}^t \phi(s)\, ds.

[edit] Proofs

The original proofs by Sudakov, Tsirelson and Borell were based on Paul Lévy's spherical isoperimetric inequality. Later, other proofs were found. In particular, Bobkov introduced a functional inequlaity that implies the G.i.i. and proved it using a certain two-point inequality. Bakry and Ledoux gave another proof of Bobkov's functuional inequality, that uses semigroup techniques and works in a much more abstract setting. Then, Barthe and Maurey gave still another proof, using the Brownian motion.

The G.i.i. also follows from Ehrhard's inequality (cf. Latała [6], Borell [7]).

[edit] 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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