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Regular version of the site

Article

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

Kolesnikov A., Milman E.

Given a probability measure μ supported on a convex subset of Euclidean

space (Rd , g0), we are interested in obtaining Poincaré and log-Sobolev type inequalities

on (, g0,μ). To this end, we change the metric g0 to a more general Riemannian one g,

adapted in a certain sense toμ, and perform our analysis on (, g,μ). The types ofmetrics we

consider are Hessian metrics (intimately related to associated optimal-transport problems),

productmetrics (which are very usefulwhenμis 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 (, g,μ) tools such as

Riemannian generalizations of theBrascamp–Lieb inequality and the Bakry–Émery criterion,

and passing back to the original Euclidean metric, we obtain various weighted inequalities on

(, g0,μ): refined and entropic versions of theBrascamp–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 themanifold (, g,μ). In some cases,we can only ensure that the lattermanifold

is (generalized) mean-convex, resulting in additional boundary terms in our inequalities.