×

zbMATH — the first resource for mathematics

Isoperimetric constants for product probability measures. (English) Zbl 0878.60013
Suppose that \((X,d)\) is a metric space equipped with a separable Borel probability measure \(\mu\), then the authors study \(\text{Is}(\mu)=\inf[\mu^+(A)/\min\{\mu(A),1- \mu(A)\}]\), the infimum is taken over all Borel sets \(A\subset X\) of measure \(0<\mu(A)<1\) and \(\mu^+(A)\) denotes the surface measure of \(A\). The main result is the inequality (1) \(\text{Is}(\mu^n)\geq (1/2\sqrt 6)\text{Is}(\mu)\), for any \(n\geq 1\), \(\mu^n\) being equal to \(\mu\otimes\cdots\otimes\mu\). An equivalent functional form (with a dimension free constant \(K(\mu)\)) is an \(L^1\)-Poincaré type inequality: (2) \(K{\mathbf E}|f|\leq{\mathbf E}|\nabla f|\), \(f\) being a function on \(\mathbb{R}^n\) with mean \({\mathbf E}f\) equal to 0, and modulus of gradient \(|\nabla f(x)|=\limsup_{y\to x} |f(y)- f(x)|/d(x,y)\). The connection between (1) and the double exponential distribution \(\nu\) on \(\mathbb{R}\) (i.e. \(\nu(dx)= 2^{-1}e^{-|x|}dx\)) is given by the following equivalent properties: (i) the probability \(\mu^n\) verifies (2) with some constant \(K\) independent of the dimension, (ii) \(\mu\) verifies (2), (iii) there exists a function \(U:\mathbb{R}\to\mathbb{R}\) with finite Lipschitz constant which transforms \(\nu\) into \(\mu\), (iv) \(\text{Is}(\mu)>0\). In addition (2) holds with \(K=\text{Is}(\mu)/(2\sqrt 6)\).
Reviewer: P.Vallois (Nancy)

MSC:
60E15 Inequalities; stochastic orderings
28A35 Measures and integrals in product spaces
49Q20 Variational problems in a geometric measure-theoretic setting
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aida, S., Masuda, T. and Shigekawa, I. (1994). Logarithmic Sobolev inequalities and exponential integrability. J. Funct. Anal. 126 83-101. · Zbl 0846.46020 · doi:10.1006/jfan.1994.1142
[2] Bakry, D. and Ledoux, M. (1996). Lévy-Gromov isoperimetric inequality for an infinite dimensional diffusion generator. Invent. Math. 123 259-286. · Zbl 0855.58011 · doi:10.1007/s002220050026 · eudml:144345
[3] Bobkov, S. G. (1997). An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probab. 25 206-214. · Zbl 0883.60031 · doi:10.1214/aop/1024404285
[4] Bobkov, S. G. and Houdré, C. (1995). Some connections between isoperimetric and Sobolevtype inequalities and isoperimetry. Memoirs AMS.
[5] Borovkov, A. A. and Utev, S. A. (1983). On an inequality and a related characterization of the normal distribution. Probab. Theory Appl. 28 219-228. · Zbl 0533.60024 · doi:10.1137/1128021
[6] Cheeger, J. (1970). A lower bound for the smallest eigenvalue of the Laplacian. In Problems in Analysis: Symposium in Honor of S. Bochner 195-199. Princeton Univ. Press. · Zbl 0212.44903 · doi:10.1007/BF02123836
[7] Gromov, M. and Milman, V. (1983). A topological application of the isoperimetric inequality. Amer. J. Math. 105 843-854. JSTOR: · Zbl 0522.53039 · doi:10.2307/2374298 · links.jstor.org
[8] Gross, L. (1993). Logarithmic Sobolev inequalities and contractivity properties of semigroups Dirichlet forms. Lecture Notes in Math. 1563 54-88. Springer, Berlin. · Zbl 0812.47037 · doi:10.1007/BFb0074091
[9] Latala, R. and Oleszkiewicz, K. (1994). On the best constant in the Khinchin-Kahane inequality. Studia Math. 109 101-104. · Zbl 0812.60010 · eudml:216056
[10] Ledoux, M. (1994a). Isoperimetry and Gaussian analysis. Ecole d’ Été de Probabilités de Saint-Flour. Lecture Notes in Math. Springer, Berlin. · Zbl 1042.60060 · doi:10.1007/s004400100161
[11] Ledoux, M. (1994b). A simple analytic proof of an inequality by P. Buser. Proc. Amer. Math. Soc. 121 951-959. · Zbl 0812.58093 · doi:10.2307/2160298
[12] Maurey, B. (1991). Some deviation inequalities. Geom. and Funct. Anal. 1 188-197. · Zbl 0756.60018 · doi:10.1007/BF01896377 · eudml:58114
[13] Muckenhoupt, B. (1972). Hardy’s inequality with weights. Studia Math. 44 31-38. · Zbl 0236.26015 · eudml:217718
[14] Pisier, G. (1986). Probabilistic methods in the geometry of Banach spaces. Probability and Analysis. Lecture Notes in Math. 1206 167-241. Springer, Berlin. · Zbl 0606.60008
[15] Talagrand, M. (1991). A new isoperimetric inequality and the concentration of measure phenomenon. Israel Seminar (GAFA). Lecture Notes in Math. 1469 94-124. Springer, Berlin. · Zbl 0818.46047 · doi:10.1007/BFb0089217
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.