Riemannian metrics on convex sets with applications to Poincar\'e and log-Sobolev inequalities

TL;DR

This paper develops new Riemannian metrics tailored to probability measures on convex sets to derive improved Poincaré and log-Sobolev inequalities, utilizing tools like curvature bounds and convexity properties.

Contribution

It introduces and analyzes Hessian, product, and conformal Riemannian metrics adapted to measures, extending inequalities in convex geometry and probability.

Findings

Derived weighted Poincaré and log-Sobolev inequalities.

Established curvature positivity conditions for the metrics.

Extended inequalities to boundary-influenced convex manifolds.

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 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.