Properties of isoperimetric, functional and Transport-Entropy inequalities via concentration

• 成果类型: Article
• 署名作者: Milman, Emanuel
• 署名单位: University of Toronto
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-010-0328-1
• 发表日期: 2012
• 页码: 475-507
• 关键词: logarithmic sobolev inequalities; log-sobolev; cost; EQUIVALENCE; convexity; PROOF; Integrability; systems; SPACES

摘要

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 (a) 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, and the dependence on the perturbation parameter is improved from linear to logarithmic. Second, the equivalence of Transport-Entropy inequalities with different cost-functions is verified, by obtaining a reverse Jensen type inequality. The main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting. Of independent interest is a new dimension independent characterization of Transport-Entropy inequalities with respect to the 1-Wasserstein distance, which does not assume any curvature lower bound.