Optimal and better transport plans

We consider the Monge–Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-cont...

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Bibliographic Details
Published in:Journal of functional analysis 2009-03, Vol.256 (6), p.1907-1927
Main Authors: Beiglböck, Mathias, Goldstern, Martin, Maresch, Gabriel, Schachermayer, Walter
Format: Article
Language:English
Subjects:
c-Cyclically monotone
Measurable cost function
Monge–Kantorovich problem
Strongly c-monotone
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Summary:We consider the Monge–Kantorovich transport problem in a purely measure theoretic setting, i.e. without imposing continuity assumptions on the cost function. It is known that transport plans which are concentrated on c-monotone sets are optimal, provided the cost function c is either lower semi-continuous and finite, or continuous and may possibly attain the value ∞. We show that this is true in a more general setting, in particular for merely Borel measurable cost functions provided that { c = ∞ } is the union of a closed set and a negligible set. In a previous paper Schachermayer and Teichmann considered strongly c-monotone transport plans and proved that every strongly c-monotone transport plan is optimal. We establish that transport plans are strongly c-monotone if and only if they satisfy a “better” notion of optimality called robust optimality.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2009.01.013