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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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