A general duality theorem for the Monge-Kantorovich transport problem

作者：Beiglboeck Mathias; Leonard Christian; Schachermayer Walter

来源：Studia Mathematica, 2012, 209(2): 151-167.

DOI：10.4064/sm209-2-4

• Monge-Kantorovich problem
• duality

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 x Y -%26gt; [0, infinity] 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 - epsilon from (X, mu) to (Y, nu) as epsilon %26gt; 0 tends to zero. %26lt;br%26gt;The classical duality theorems of H. Kellerer, where c is lower semicontinuous or uniformly bounded, quickly follow from these general results.

• 出版日期2012