Cyclical monotonicity and the ergodic theorem

Abstract

AbstractIt is well known that optimal transport plans are cyclically monotone. The reverse implication that cyclically monotone transport plans are optimal needs some assumptions and the proof is non-trivial even if the costs are given by the squared euclidean distance on ${ \mathbb{R} }^{n} $. We establish this result as a corollary to the ergodic theorem.