×

zbMATH — the first resource for mathematics

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

MSC:
60G15 Gaussian processes
60E15 Inequalities; stochastic orderings
60B05 Probability measures on topological spaces
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] BAKRY, D. and LEDOUX, M. 1995. Levy Gromov isoperimetric inequality for an infinite \' dimensional diffusion generator. Invent. Math. 123 259 281. · Zbl 0855.58011
[2] BOBKOV, S. 1996. A functional form of the isoperimetric inequality for the Gaussian measure. J. Funct. Anal. 135 39 49. · Zbl 0838.60013
[3] BORELL, C. 1975. The Brunn Minkowski inequality in Gauss space. Invent. Math. 30 207 216. · Zbl 0311.60007
[4] EHRHARD, A. 1983. Symetrisation dans l’espace de Gauss. Math. Scand. 53 281 301. \' · Zbl 0542.60003
[5] GROSS, L. 1993. Logarithmic Sobolev inequalities and contractivity properties of semigroups. Varenna 1992. Lecture Notes in Math. 1563 54 88. Springer, Berlin. · Zbl 0812.47037
[6] LEDOUX, M. 1994. Isoperimetry and Gaussian analysis. Ecole d’Ete de Probabilites de Saint \' \' Flour. Lecture Notes in Math. Springer, Berlin. · Zbl 0874.60005
[7] SUDAKOV, V. N. and TSIREL’SON, B. S. 1978. Extremal properties of half-spaces for spheri cally invariant measures. J. Soviet Math. 9 9 18. Translated from Zap. Nauchn.Sem. Leningrad. Otdel. Math. Inst. Steklova. 41 1974 14 24. · Zbl 0395.28007
[8] TALAGRAND, M. 1993. Isoperimetry, logarithmic Sobolev inequalities on the discrete cube, and Margulis’ graph connectivity theorem. Geom. and Funct. Anal. 3 295 314. · Zbl 0806.46035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.