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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## A general duality theorem for the Monge–Kantorovich transport problem

### Tom 209 / 2012

Studia Mathematica 209 (2012), 151-167 MSC: Primary 49Q20. DOI: 10.4064/sm209-2-4

#### Streszczenie

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.

#### Autorzy

• 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

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