Blaschke–Santaló 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 Blaschke–Santaló inequality and affine isoperimetric inequalities for many sets functions. We derive it an entropy bound total Kantorovich cost appearing in problem. also establish “pointwise Prékopa–Leindler inequality” show monotonicity property multimarginal Blaschke–Santaó functional.