# Gaussian isoperimetric inequality

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 $A_\varepsilon = \left\{ x \in \mathbf{R}^n \, | \, \text{dist}(x, A) \leq \varepsilon \right\}$
$\liminf_{\varepsilon \to +0} \varepsilon^{-1} \left\{ \gamma^n (A_\varepsilon) - \gamma^n(A) \right\} \geq \varphi(\Phi^{-1}(\gamma^n(A))),$
$\varphi(t) = \frac{\exp(-t^2/2)}{\sqrt{2\pi}}\quad{\rm and}\quad\Phi(t) = \int_{-\infty}^t \varphi(s)\, ds.$