Properties of Isoperimetric, Functional and Transport-Entropy Inequalities Via Concentration TeX Export

@article{citeulike:5705272, abstract = {Various properties of isoperimetric, functional, Transport-Entropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a one-sided \$L^\infty\$ bound on the ratio between their densities, Wasserstein distances, and Kullback - Leibler divergence. In particular, an extension of the Holley--Stroock perturbation lemma for the log-Sobolev inequality is obtained. Second, the equivalence of Transport-Entropy inequalities with different cost functions is verified, by obtaining a reverse Jensen type inequality. In view of a recent result of Gozlan, this is used to obtain tensorization properties of concentration inequalities with respect to various product-metrics, and the tensorization result for isoperimetric inequalities of Barthe--Cattiaux--Roberto is easily recovered. Some further applications are also described. The main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting.}, archivePrefix = {arXiv}, author = {Milman, Emanuel}, citeulike-article-id = {5705272}, citeulike-linkout-0 = {http://arxiv.org/abs/0909.0207}, citeulike-linkout-1 = {http://arxiv.org/pdf/0909.0207}, day = {1}, eprint = {0909.0207}, keywords = {inequality, transport}, month = {Sep}, posted-at = {2009-09-02 07:29:03}, priority = {2}, title = {Properties of Isoperimetric, Functional and Transport-Entropy Inequalities Via Concentration}, url = {http://arxiv.org/abs/0909.0207}, year = {2009} }

TY - JOUR ID - citeulike:5705272 L3 - citeulike-article-id:5705272 N2 - Various properties of isoperimetric, functional, Transport-Entropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a one-sided $L^\infty$ bound on the ratio between their densities, Wasserstein distances, and Kullback - Leibler divergence. In particular, an extension of the Holley--Stroock perturbation lemma for the log-Sobolev inequality is obtained. Second, the equivalence of Transport-Entropy inequalities with different cost functions is verified, by obtaining a reverse Jensen type inequality. In view of a recent result of Gozlan, this is used to obtain tensorization properties of concentration inequalities with respect to various product-metrics, and the tensorization result for isoperimetric inequalities of Barthe--Cattiaux--Roberto is easily recovered. Some further applications are also described. The main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting. TI - Properties of Isoperimetric, Functional and Transport-Entropy Inequalities Via Concentration KW - inequality KW - transport AU - Milman, Emanuel PY - 2009/Sep/01/ UR - http://arxiv.org/abs/0909.0207 ER -