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

Studia Mathematica (2012)

• Volume: 209, Issue: 2, page 151-167
• ISSN: 0039-3223

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## Abstract

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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 μ and ν. The transport cost function c: X × Y → [0,∞] 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 - ε from (X,μ) to (Y,ν) as ε > 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.

## How to cite

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Mathias Beiglböck, Christian Léonard, and Walter Schachermayer. "A general duality theorem for the Monge-Kantorovich transport problem." Studia Mathematica 209.2 (2012): 151-167. <http://eudml.org/doc/285844>.

@article{MathiasBeiglböck2012,
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 μ and ν. The transport cost function c: X × Y → [0,∞] 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 - ε from (X,μ) to (Y,ν) as ε > 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. },
author = {Mathias Beiglböck, Christian Léonard, Walter Schachermayer},
journal = {Studia Mathematica},
keywords = {Monge-Kantorovich problem; duality; Borel probability measures; no duality gap; relaxed transport problem},
language = {eng},
number = {2},
pages = {151-167},
title = {A general duality theorem for the Monge-Kantorovich transport problem},
url = {http://eudml.org/doc/285844},
volume = {209},
year = {2012},
}

TY - JOUR
AU - Mathias Beiglböck
AU - Christian Léonard
AU - Walter Schachermayer
TI - A general duality theorem for the Monge-Kantorovich transport problem
JO - Studia Mathematica
PY - 2012
VL - 209
IS - 2
SP - 151
EP - 167
AB - 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 μ and ν. The transport cost function c: X × Y → [0,∞] 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 - ε from (X,μ) to (Y,ν) as ε > 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.
LA - eng
KW - Monge-Kantorovich problem; duality; Borel probability measures; no duality gap; relaxed transport problem
UR - http://eudml.org/doc/285844
ER -

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