Isoperimetric constants for product probability measures. (English) Zbl 0878.60013

Bobkov, S. G.; Houdré, C.

Ann. Probab. 25, No. 1, 184-205 (1997).

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

Keywords:

\(L^ 1\)-Poincaré type inequality; double exponential distribution

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