Properties of Isoperimetric, Functional and Transport-Entropy Inequalities Via Concentration - Mathematics > Functional AnalysisReport as inadecuate




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Abstract: Various properties of isoperimetric, functional, Transport-Entropy andconcentration inequalities are studied on a Riemannian manifold equipped with ameasure, whose generalized Ricci curvature is bounded from below. First,stability of these inequalities with respect to perturbation of the measure isobtained. The extent of the perturbation is measured using several differentdistances 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 theHolley-Stroock perturbation lemma for the log-Sobolev inequality is obtained,and the dependence on the perturbation parameter is improved from linear tologarithmic. Second, the equivalence of Transport-Entropy inequalities withdifferent cost-functions is verified, by obtaining a reverse Jensen typeinequality. The main tool used is a previous precise result on the equivalencebetween concentration and isoperimetric inequalities in the described setting.Of independent interest is a new dimension independent characterization ofTransport-Entropy inequalities with respect to the 1-Wasserstein distance,which does not assume any curvature lower bound.



Author: Emanuel Milman

Source: https://arxiv.org/



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