Gaussian isoperimetric inequality

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

Equivalently,


 * $$\liminf_{\varepsilon \to +0}

\varepsilon^{-1} \left\{ \gamma^n (A_\varepsilon) - \gamma^n(A) \right\} \geq \varphi(\Phi^{-1}(\gamma^n(A))),$$

where &gamma;n is a Gaussian measure on $$\scriptstyle\mathbf{R}^n$$,


 * $$A_\varepsilon = \left\{ x \in \mathbf{R}^n \, | \,

\text{dist}(x, A) \leq \varepsilon \right\}$$

is the &epsilon;-extension of A,


 * $$\varphi(t) = \frac{\exp(-t^2/2)}{\sqrt{2\pi}}$$

and


 * $$\Phi(t) = \int_{-\infty}^t \varphi(s)\, ds. $$

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]).