A general duality theorem for the Monge–Kantorovich transport problem
Volume 209 / 2012
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 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.
