Blaschke-Santalo Inequality for Many Functions and Geodesic Barycenters of Measures

Explore a 56-minute lecture on the generalization of the Blaschke-Santalo inequality and its applications to geodesic barycenters in optimal transportation theory. Delve into the work of Alex Kolesnikov and Elisabeth Werner as they present a natural extension of this inequality for multiple sets and functions. Discover how this generalization leads to an entropy bound for the total Kantorovich cost in the barycenter problem. Examine key concepts such as body centers of probability measures, inverse mappings, and transportational inequalities. Investigate the pointwise inequality, dual convex body, and the standard proof approach. Learn about the inequality for Gaussian measures and the transportation proof method. Gain insights into the duality relation, monotonicity, and the Keller-Einstein collision concept as they relate to this mathematical exploration.