Blaschke-Santalo inequality for many functions and geodesic barycenters of measures

Motivated by the geodesic barycenter problem from optimal transportation theory, we prove a natural generalization of the Blaschke-Santalo inequality and the affine isoperimetric inequalities for many sets and many functions. We derive from it an entropy bound for the total Kantorovich cost appearing in the barycenter problem. We also establish a pointwise Prekopa-Leindler inequality and show a monotonicity property of the multimarginal Blaschke-Santalo functional.