Homogeneous maximizers of the Blaschke--Santalo-type functionals

TL;DR

This paper extends Blaschke--Santal{'}o}-type inequalities to multiple sets and special cost functions, introducing reduction techniques and symmetrization methods, with links to optimal transportation and information inequalities.

Contribution

It introduces new reduction methods for maximizing Blaschke--Santal{'}o}-type functionals for multiple sets and extends symmetrization techniques beyond two sets.

Findings

Reduction of the maximization problem to homogeneous cases.

Extension of symmetrization to more than two sets.

Connections to multimarginal optimal transportation and sharp inequalities.

Abstract

We study Blaschke--Santal{\'o}-type inequalities for $N \ge 2$ sets (functions) and a special class of cost functions. In particular, we prove new results about reduction of the maximization problem for the Blaschke--Santal{\'o}-type functional to homogeneous case (functional inequalities on the sphere) and extend the symmetrization argument to the case of $N > 2$ sets. We also discuss links to the multimagrinal optimal transportation problem and the related sharp transportation-information inequalities.