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

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