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Blaschke–Santaló inequality for many functions and geodesic barycenters of measures
Advances in Mathematics. 2022. Vol. 396. Article 108110.
Kolesnikov A., Werner E.
Motivated by the geodesic barycenter problem from optimal transportation theory, we prove a natural generalization of the Blaschke–Santaló 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 Prékopa–Leindler inequality” and show a monotonicity property of the multimarginal Blaschke–Santaó functional.
Keywords: Monge-Kantorovich problemзадача Монжа-КанторовичаWasserstein barycenterBlaschke-Santalo inequalityбарицентры мернеравенство Бляшке-Сантало
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