A+ CATEGORY SCIENTIFIC UNIT

A general duality theorem for the Monge–Kantorovich transport problem

Volume 209 / 2012

Mathias Beiglböck, Christian Léonard, Walter Schachermayer Studia Mathematica 209 (2012), 151-167 MSC: Primary 49Q20. DOI: 10.4064/sm209-2-4

Abstract

The duality theory for 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 semicontinuous or uniformly bounded, quickly follow from these general results.

Authors

  • Mathias BeiglböckFaculty of Mathematics
    University of Vienna
    Nordbergstrasse 15
    1090 Wien, Austria
    e-mail
  • Christian LéonardModal-X, Université Paris Ouest
    Bât. G, 200 av. de la République
    92001 Nanterre, France
    e-mail
  • Walter SchachermayerFaculty of Mathematics
    University of Vienna
    Nordbergstrasse 15
    1090 Wien, Austria
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image