Gaussian Isoperimetric Inequality

Theorem
Let $A$ be a measurable subset of $\R^n$ endowed with the standard Gaussian measure $\gamma^n$ with the density $\dfrac {\map \exp {-\norm x^2 / 2} } {\paren {2 \pi}^{n/2} }$

Denote by
 * $A_\epsilon = \set {x \in \R^n : \map {\operatorname {dist} } {x, A} \le \epsilon}$

the $\epsilon$-extension of $A$.

The Gaussian isoperimetric inequality states that:


 * $\displaystyle \liminf_{\epsilon \mathop \to +0} \epsilon^{-1} \set {\map {\gamma^n} {A_\epsilon} - \map {\gamma^n} A} \ge \map \varphi {\map {\Phi^{-1} } {\map {\gamma^n} A } }$

where:


 * $\map \varphi t = \dfrac {\map \exp {-t^2 / 2} } {\sqrt {2 \pi} }$

and:
 * $\map \Phi t = \displaystyle \int_{-\infty}^t \map \varphi s \rd s$