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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
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##### References:
  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  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  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  Bobkov, S. G. and Houdré, C. (1995). Some connections between isoperimetric and Sobolevtype inequalities and isoperimetry. Memoirs AMS.  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  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  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  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  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  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  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  Maurey, B. (1991). Some deviation inequalities. Geom. and Funct. Anal. 1 188-197. · Zbl 0756.60018 · doi:10.1007/BF01896377 · eudml:58114  Muckenhoupt, B. (1972). Hardy’s inequality with weights. Studia Math. 44 31-38. · Zbl 0236.26015 · eudml:217718  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  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
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