Matrix H\"older's inequality and divergence formulation of optimal transport of vector measures

TL;DR

This paper characterizes equality in matrix Hölder's inequality and introduces a divergence-based approach to optimal transport of vector measures, extending classical results to broader polar cone settings.

Contribution

It provides a new divergence formulation for optimal transport of vector measures and generalizes representation formulas to various polar cones.

Findings

Characterization of equality cases in matrix Hölder's inequality

Divergence formulation for optimal transport of vector measures

Extension of representation formulas to multiple polar cones

Abstract

We characterise equality cases in matrix H\"older's inequality and develop a divergence formulation of optimal transport of vector measures. As an application, we reprove the representation formula for measures in the polar cone to monotone maps. We generalise the last result to a wide class of polar cones, including polar cones to tangent cones to the unit ball in the space of differentiable functions and in the Sobolev spaces.