A Riemannian interpolation inequality a la Borell, Brascamp and Lieb

A concavity estimate is derived for interpolations between L-1 (M) mass den sities on a Riemannian manifold. The inequality sheds new light on the theo rems of Prekopa, Leindler, Borell, Brascamp and Lieb that it generalizes fr om Euclidean space. Due to the curvature of the manifold, the new Riemannia n versions of these theorems incorporate a volume distortion factor which c an, however, be controlled via lower bounds on Ricci curvature. The method uses optimal mappings from mass transportation theory. Along the way, sever al new properties are established for optimal mass transport and interpolat ing maps on a Riemannian manifold.