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

TL;DR

This paper generalizes classical inequalities to multiple functions and sets within the context of optimal transportation, providing new bounds and properties relevant to geodesic barycenters and entropy.

Contribution

It introduces a generalized Blaschke-Santalo inequality for many functions and sets, and derives entropy bounds for Kantorovich costs in barycenter problems.

Findings

Generalized Blaschke-Santalo inequality for multiple functions and sets

Established a pointwise Prekopa-Leindler inequality

Proved a monotonicity property of the multimarginal Blaschke-Santalo functional

Abstract

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.