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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 $(\mathbb{R}^d,g_0)$, we are interested in obtaining Poincaré and log-Sobolev type inequalities on $(Ω,g_0,μ)$. To this end, we change the metric $g_0$ 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 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 $(Ω,g,μ)$ 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 $(Ω,g_0,μ)$: 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.