  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.
