Calculus of Variations and Geometric Measure Theory

M. O. Rieger

Monotonicity of transport plans

created by rieger on 29 Sep 2006


Submitted Paper

Inserted: 29 sep 2006

Year: 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