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Optimal transportation, topology and uniqueness. (English) Zbl 1255.49075
Summary: The Monge-Kantorovich transportation problem involves optimizing with respect to a given cost function. Uniqueness is a fundamental open question about which is only little known if the cost function is smooth and the landscapes containing the goods to be transported possess (non-trivial) topology. This question turns out to be closely linked to a delicate problem (111) of [G. Birkhoff, Lattice Theory. Rev. ed. American Mathematical Society Colloquium Publications 25. New York: American Mathematical Society (AMS) (1948; Zbl 0033.10103)]: give a necessary and sufficient condition on the support of a joint probability to guarantee extremality among all measures which share its marginals. Fifty years of progress on Birkhoff’s question culminate in Hestir and Williams’ necessary condition which is nearly sufficient for extremality; we relax their subtle measurability hypotheses separating necessity from sufficiency slightly, yet demonstrate by example that to be sufficient certainly requires some measurability. Their condition amounts to the vanishing of the measure \(\gamma\) outside a countable alternating sequence of graphs and antigraphs in which no two graphs (or two antigraphs) have domains that overlap, and where the domain of each graph/antigraph in the sequence contains the range of the succeeding antigraph (respectively, graph). Such sequences are called numbered limb systems. We then explain how this characterization can be used to resolve the uniqueness of Kantorovich solutions for optimal transportation on a manifold with the topology of the sphere.
Reviewer: Reviewer (Berlin)

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
93E20 Optimal stochastic control
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