Matrix Hölder's inequality and divergence formulation of optimal transport of vector measures

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.