An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space
The author finds an elementary method of proving isoperimetric inequalities for the canonical Gaussian measure \(\gamma _n\) on \(\mathbb{R}^n\). In particular, the method permits to prove that for any Borel set \(A\subset\mathbb{R}^n\), \(\gamma_n^+(A)\geq \varphi (\Phi^{-1}(\gamma_n(A))\), where \(\gamma_n^+\) is the Minkowski surface measure with respect to \(\gamma_n\), \(\Phi \) is the distribution function of a standard Gaussian random variable, while \(\varphi \) is its density function. As a corollary the following well-known original form of isoperimetric inequality is derived: For any Borel \(A\subset\mathbb{R}^n \) and any \(h>0\), \(\gamma_n(A^h) \geq \Phi (\Phi^{-1}(\gamma_n(A)) + h)\), where \(A^h\) stands for the open \(h\)-neighborhood of \(A\).
scientific article; zbMATH DE number 1018387
scientific article