Abstract
Chapter 1 gave, in the framework of the general theory of optimal transportation based on duality methods, an existence result for the optimal transport map when the cost is of the form \(c(x,y) = \vert x - y\vert ^{p}\), for \(p \in ]1,+\infty [\). We look in this chapter at the two limit cases p = 1 and \(p = \infty \), which require additional techniques. We prove existence of an optimal map under absolutely continuous assumptions on the source measure. Then, the discussion section will go into two different directions: on the one hand the L 1 and \(L^{\infty }\) cases introduced and motivated the study of convex costs which could be non strictly-convex or infinite-valued somewhere; on the other hand one could wonder what is the situation for p < 1, i.e. for costs which are concave increasing functions of the distance.
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Notes
- 1.
Measurability could be proven, either by restricting to a \(\sigma\)-compact set Γ or by considering the disintegrations μ s and ν s and using the fact that, on each s, T is the monotone map sending μ s onto ν s (and hence it inherits some measurability properties of the dependence of μ s and ν s w.r.t. s, which are guaranteed by abstract disintegration theorems).
- 2.
Note that the construction is essentially the same as in the example provided in [199], for a different goal. The regularity degree is slightly different, and we decided to handle by hands “vertical” Lipschitz curves in order to make a self-contained presentation.
- 3.
We mean here costs which are concave functions of the distance | x − y | , not of the displacement x − y, as instead we considered in Remark 2.12.
- 4.
We will not explicitly state it every time, but this also implies that ℓ is strictly increasing.
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Santambrogio, F. (2015). L 1 and L ∞ theory. In: Optimal Transport for Applied Mathematicians. Progress in Nonlinear Differential Equations and Their Applications, vol 87. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-20828-2_3
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