A generalized dual maximizer for the Monge--Kantorovich transport problem

Mathias Beiglböck; Christian Léonard; Walter Schachermayer

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arXiv · 1009.1118

A generalized dual maximizer for the Monge--Kantorovich transport problem

Mathias Beiglböck·Christian Léonard·Walter Schachermayer

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Abstract

The dual attainment 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 $c:\XY \to [0,\infty]$ is assumed to be Borel measurable. We show that a dual optimizer always exists, provided we interpret it as a projective limit of certain finitely additive measures. Our methods are functional analytic and rely on Fenchel's perturbation technique.

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Mathias Beiglböck, Christian Léonard, Walter Schachermayer. 2010-09-06. A generalized dual maximizer for the Monge--Kantorovich transport problem. https://arxiv.org/abs/1009.1118

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