Gaussian isoperimetric inequality

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

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

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).

References

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.

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Gaussian isoperimetric inequality

Mathematical formulation

Remarks on the proofs

See also

References