Inserted: 29 sep 2006
We study monotonicity properties for minimizers of transport problems. In the one-dimensional case, we present an algorithm to construct minimizing monotone transport plans by ``monotonizing'' a given minimizing transport plan. This method applies in particular to the case of the $L^1$-Wasserstein metric where we prove the existence of monotone minimizers for arbitrary marginals. We find that monotone transport plans are in a certain sense close to monotone transport maps.
Keywords: Wasserstein distance, cyclical monotonicity, transport problems, covariance