Geodesic
Transport problems and disintegration maps
ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 888-905.

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By disintegration of transport plans it is introduced the notion of transport class. This allows to consider the Monge problem as a particular case of the Kantorovich transport problem, once a transport class is fixed. The transport problem constrained to a fixed transport class is equivalent to an abstract Monge problem over a Wasserstein space of probability measures. Concerning solvability of this kind of constrained problems, it turns out that in some sense the Monge problem corresponds to a lucky case.

DOI : 10.1051/cocv/2012037
Classification : 37J50, 49Q20, 49Q15
Keywords: optimal mass transportation theory, Monge − Kantorovich problem, calculus of variations, shape analysis, geometric measure theory 
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Granieri, Luca; Maddalena, Francesco. Transport problems and disintegration maps. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 3, pp. 888-905. doi : 10.1051/cocv/2012037. http://geodesic.mathdoc.fr/articles/10.1051/cocv/2012037/

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