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

Given a probability measure μ supported on a convex subset Ω of Euclidean space (R^d, g₀) , we are interested in obtaining Poincaré and log-Sobolev type inequalities on (Ω , g₀, μ). To this end, we change the metric g₀ to a more general Riemannian one g, adapted in a certain sense to μ, and perform our analysis on (Ω , g, μ). The types of metrics we consider are Hessian metrics (intimately related to associated optimal-transport problems), product metrics (which are very useful when μ is unconditional, i.e. invariant under reflections with respect to the coordinate hyperplanes), and metrics conformal to the Euclidean one, which have not been previously explored in this context. Invoking on (Ω , g, μ) tools such as Riemannian generalizations of the Brascamp–Lieb inequality and the Bakry–Émery criterion, and passing back to the original Euclidean metric, we obtain various weighted inequalities on (Ω , g₀, μ) : refined and entropic versions of the Brascamp–Lieb inequality, weighted Poincaré and log-Sobolev inequalities, Hardy-type inequalities, etc. Key to our analysis is the positivity of the associated Lichnerowicz–Bakry–Émery generalized Ricci curvature tensor, and the convexity of the manifold (Ω , g, μ). In some cases, we can only ensure that the latter manifold is (generalized) mean-convex, resulting in additional boundary terms in our inequalities.