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AN EXISTENCE RESULT FOR THE MONGE PROBLEM IN R^n WITH NORM COST FUNCTIONS
, 2009
"... We establish existence of solutions to the Monge problem in n with a norm cost function, assuming absolute continuity of the initial measure. The loss in strict convexity of the unit ball implies that transport is possible along several directions. As in [4], we single out particular solutions to t ..."
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We establish existence of solutions to the Monge problem in n with a norm cost function, assuming absolute continuity of the initial measure. The loss in strict convexity of the unit ball implies that transport is possible along several directions. As in [4], we single out particular solutions to the Kantorovich relaxation with a secondary variational problem, which involves a strictly convex norm. We then define a map rearranging the mass within the rays, by a Sudakovtype argument with the disintegration technique in [8, 10, 17]. In the secondary variational problem the cost is also infinite valued and in general there is no Kantorovich potential. However, all the optimal transport plans share the same maximal transport rays. We derive also an expression for the transport density associated to these optimal plans. Remark: The construction presently given in the preprint needs the further technical assumption that, with the notation of Section 3.4, the set of points x ∈ Ts whose secondary transport ray belongs to the
A STRATEGY FOR NONSTRICTLY CONVEX TRANSPORT COSTS AND THE EXAMPLE OF x − y^p IN R²
, 2009
"... This paper deals with the existence of optimal transport maps for some optimal transport problems with a convex but non strictly convex cost. We give a decomposition strategy to address this issue. As a consequence of our procedure, we have to treat some transport problems, of independent interest, ..."
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This paper deals with the existence of optimal transport maps for some optimal transport problems with a convex but non strictly convex cost. We give a decomposition strategy to address this issue. As a consequence of our procedure, we have to treat some transport problems, of independent interest, with a convex constraint on the displacement. To illustrate possible results obtained through this general approach, we prove existence of optimal transport maps in the case where the source measure is absolutely continuous with respect to the Lebesgue measure and the transportation cost is of the form h(‖x − y‖) with h strictly convex increasing and ‖. ‖ an arbitrary norm in R².
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"... sous la coordination de Ivar Ekeland Problèmes classiques et moins classiques en transport optimal Régularité, approximation, EDP et applications ..."
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sous la coordination de Ivar Ekeland Problèmes classiques et moins classiques en transport optimal Régularité, approximation, EDP et applications