A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. (English) Zbl 1026.58018

Cordero-Erausquin, Dario; McCann, Robert J.; Schmuckenschläger, Michael

Invent. Math. 146, No. 2, 219-257 (2001).

The aim of this paper is to extend important functional inequalities from the Euclidean case to an \(n\)-dimensional complete, connected, Riemannian \(C^2\)-manifold. These inequalities are interpolation inequalities. The main result is a Riemannian Borell-Brascamp-Lieb inequality. It has as significant corollaries various Riemannian \(p\)-mean inequalities. For instance, for \(p=0\) one obtains a Riemannian version of the Prékopa-Leindler inequality. The method of proof relies on the study of optimal mass transport.

Reviewer: Dumitru Motreanu (Perpignan)

MSC:

58E35	Variational inequalities (global problems) in infinite-dimensional spaces
28C99	Set functions and measures on spaces with additional structure
60E15	Inequalities; stochastic orderings

Keywords:

interpolation inequalities; Riemannian manifold; curvature; goedesics; optimal mass transport