Abstract
Poincare-type estimates for a logarithmically concave measure μ on a convex set Ω are obtained. For this purpose, Ω is endowed with a Riemannian metric g in which the Riemannian manifold with measure (Ω, g, μ) has nonnegative Bakry–Emery tensor and, as a corollary, satisfies the Brascamp–Lieb inequality. Several natural classes of metrics (such as Hessian and conformal metrics) are considered; each of these metrics gives new weighted Poincare, Hardy, or log-Sobolev type inequalities and other results.
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