The Monge--Kantorovich problem, the Schur--Horn theorem, and the diffeomorphism group of the annulus

First, we analyze the discrete Monge--Kantorovich problem, linking it with the minimization problem of linear functionals over adjoint orbits. Second, we consider its generalization to the setting of area preserving diffeomorphisms of the annulus. In both cases, we show how the problem can be linked to permutohedra, majorization, and to gradient flows with respect to a suitable metric.