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 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

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

References