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

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