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

Abstract

Given a probability measure $\mu$ supported on a convex subset $\Omega$ of Euclidean space $({\mathbb{R}}^d,g_0)$, we are interested in obtaining Poincaré 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--Émery 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 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 $(\Omega ,g,\mu )$. In some cases, we can only ensure that the latter manifold is (generalized) mean-convex, resulting in additional boundary terms in our inequalities.

46 pages, to appear in Calc. Var. & PDE

References in corpus (3)

• Weighted Poincaré-type inequalities for Cauchy and other convex measures
• An isoperimetric inequality on the ${\ell }_p$ balls
• Eigenvalue distribution of optimal transportation

Cited by in corpus (3)

• Stein kernels and moment maps
• Poincaré and log-Sobolev inequalities for mixtures
• The Brownian transport map