Gaussian Isoperimetric Inequality

Statement
Let $$\scriptstyle A$$ be a measurable subset of $$\scriptstyle\mathbf{R}^n $$ endowed with the standard Gaussian measure $$\gamma^n$$ with the density $$ {\exp(-\|x\|^2/2)}/(2\pi)^{n/2}$$. Denote by
 * $$A_\varepsilon = \left\{ x \in \mathbf{R}^n \, | \,

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

the &epsilon;-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


 * $$\varphi(t) = \frac{\exp(-t^2/2)}{\sqrt{2\pi}}\quad{\rm and}\quad\Phi(t) = \int_{-\infty}^t \varphi(s)\, ds. $$