An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space. (English) Zbl 0883.60031

Bobkov, S. G.

Ann. Probab. 25, No. 1, 206-214 (1997).

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 \(A\).

Reviewer: S.A.Chobanjan (Tbilisi)

MSC:

60G15	Gaussian processes
60E15	Inequalities; stochastic orderings
60B05	Probability measures on topological spaces

Keywords:

isoperimetric inequality; Gaussian measure; discrete cube

References:

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