Gaussian isoperimetric inequality

From ProofWiki
Jump to: navigation, search

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 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

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