Gaussian isoperimetric inequality

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In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov and independently by Christer Borell, 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[edit]

Let \scriptstyle A be a measurable subset of \scriptstyle\mathbf{R}^n endowed with the Gaussian measure γ n. 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.

Remarks on the proofs[edit]

The original proofs by Sudakov, Tsirelson and Borell were based on Paul Lévy's spherical 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 the semigroup techniques which works in a much more abstract setting. Later Barthe and Maurey gave yet another proof using the Brownian motion.

The Gaussian isoperimetric inequality also follows from Ehrhard's inequality (cf. Latała, Borell).

See also[edit]

References[edit]

  • 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
  • Ch. Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30 (1975), no. 2, 207–216.
  • 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
  • D.Bakry, M.Ledoux, Lévy-Gromov's isoperimetric inequality for an infinite-dimensional diffusion generator, Invent. Math. 123 (1996), no. 2, 259–281
  • F. Barthe, B. Maurey, Some remarks on isoperimetry of Gaussian type, Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 4, 419–434.
  • R. Latała, A note on the Ehrhard inequality, Studia Math. 118 (1996), no. 2, 169–174.
  • Ch. Borell, The Ehrhard inequality, C. R. Math. Acad. Sci. Paris 337 (2003), no. 10, 663–666.