A General Duality Theorem for the Monge--Kantorovich Transport Problem

Mathias Beiglboeck; Christian Leonard; Walter Schachermayer

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arXiv · 0911.4347

A General Duality Theorem for the Monge--Kantorovich Transport Problem

Mathias Beiglboeck·Christian Leonard·Walter Schachermayer

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Abstract

The duality theory of the Monge--Kantorovich transport problem is analyzed in a general setting. The spaces $X, Y$ are assumed to be polish and equipped with Borel probability measures $\mu$ and $\nu$. The transport cost function $c:X\times Y \to [0,\infty]$ is assumed to be Borel. Our main result states that in this setting there is no duality gap, provided the optimal transport problem is formulated in a suitably relaxed way. The relaxed transport problem is defined as the limiting cost of the partial transport of masses $1-\varepsilon$ from $(X,\mu)$ to $(Y, \nu)$, as $\varepsilon >0$ tends to zero. The classical duality theorems of H.\ Kellerer, where $c$ is lower semi-continuous or uniformly bounded, quickly follow from these general results.

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Mathias Beiglboeck, Christian Leonard, Walter Schachermayer. 2009-11-23. A General Duality Theorem for the Monge--Kantorovich Transport Problem. https://arxiv.org/abs/0911.4347

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