Full characterization of optimal transport plans for concave costs. (English) Zbl 1334.49146

Piazzoli, Davide; Santambrogio, Filippo; Pegon, Paul

Discrete Contin. Dyn. Syst. 35, No. 12, 6113-6132 (2015).

Summary: This paper slightly improves a classical result by Gangbo and McCann from 1996 about the structure of optimal transport plans for costs that are strictly concave and increasing functions

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of the Euclidean distance

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. Since the main difficulty for proving the existence of an optimal map comes from the possible singularity of the cost at \(0\), everything is quite easy if the supports of the two measures are disjoint; Gangbo and McCann proved the result under the assumption \(\mu(\mathrm{supp}(\nu))=0\); in this paper we replace this assumption with the fact that the two measures are singular to each other. In this case it is possible to prove the existence of an optimal transport map, provided the starting measure \(\mu\) does not give mass to small sets

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(i.e. \((d-1)\)-rectifiable sets

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). When the measures are not singular, the optimal transport plan decomposes into two parts, one concentrated on the diagonal and the other being a transport map between mutually singular measures

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.

MSC:

49Q20	Variational problems in a geometric measure-theoretic setting
49J45	Methods involving semicontinuity and convergence; relaxation
49K21	Optimality conditions for problems involving relations other than differential equations
28A75	Length, area, volume, other geometric measure theory

Keywords:

optimal transport plans; concave costs; approximate gradient; rectifiable sets; density points

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