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

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.

MSC:

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