Relative optimal transport

We develop a theory of optimal transport relative to a distinguished subset, which acts as a reservoir of mass, allowing us to compare measures of different total variation. This relative transportation problem has an optimal solution and we obtain relative versions of the Kantorovich–Rubinstein norm, Wasserstein distance, Kantorovich–Rubinstein duality and Monge–Kantorovich duality. We also prove relative versions of the Riesz–Markov–Kakutani theorem, which connect the spaces of measures arising from the relative optimal transport problem to spaces of Lipschitz functions. For a boundedly compact Polish space, we show that our relative 1-finite real-valued Radon measures with relative Kantorovich–Rubinstein norm coincide with the sequentially order continuous dual of relative Lipschitz functions with the operator norm. As part of our work we develop a theory of Riesz cones that may be of independent interest.