A concavity estimate is derived for interpolations between L 1 (M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian versions of these theorems incorporate a volume distortion factor which can, however, be controlled via lower bounds on Ricci curvature. The method uses optimal mappings from mass transportation theory. Along the way, several new properties are established for optimal mass transport and interpolating maps on a Riemannian manifold.

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We show that given a symmetric convex set KCR d ; the function t-gðe t KÞ is log-concave on R; where g denotes the standard d-dimensional Gaussian measure. We also comment on the extension of this property to unconditional log-concave measures and sets, and on the complex case. r

As discovered by Brenier, mapping through a convex gradient gives the optimal transport in R n . In the present article, this map is used in the setting of Gaussianlike measures to derive an inequality linking entropy with mass displacement by a straightforward argument. As a consequence, logarithmic Sobolev and transport inequalities are recovered. Finally, a result of Caffarelli on the Brenier map is used to obtain Gaussian correlation inequalities.