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Working paper

Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities.

Kolesnikov A., Milman E.
Given a probability measure \mu supported on a convex subset \Omega of Euclidean space (\mathbb{R}^d,g_0), we are interested in obtaining Poincar\'e and log-Sobolev type inequalities on (\Omega,g_0,\mu). To this end, we change the metric g_0 to a more general Riemannian one g, adapted in a certain sense to \mu, and perform our analysis on (\Omega,g,\mu). The types of metrics we consider are Hessian metrics (intimately related to associated optimal-transport problems), product metrics (which are very useful when \mu is unconditional, i.e. invariant under reflection with respect to the principle hyperplanes), and metrics conformal to the Euclidean one, which have not been previously explored in this context. Invoking on (\Omega,g,\mu) tools such as Riemannian generalizations of the Brascamp--Lieb inequality and the Bakry--\'Emery criterion, and passing back to the original Euclidean metric, we obtain various weighted inequalities on(\Omega,g_0,\mu): refined and entropic versions of the Brascamp--Lieb inequality, weighted Poincar\'e and log-Sobolev inequalities, Hardy-type inequalities, etc. Key to our analysis is the positivity of the associated Lichnerowicz--Bakry--\'Emery generalized Ricci curvature tensor, and the convexity of the manifold(\Omega,g,\mu). In some cases, we can only ensure that the latter manifold is (generalized) mean-convex, resulting in additional boundary terms in our inequalities.