Optimal transportation, topology and uniqueness

By N. Ahmad, Hwa Kil Kim, Robert J. McCann. 2010

49 citations recorded by the source corpus.

The Monge–Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when 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 Birkhoff (Lattice Theory. Revised Edition, 1948): 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 γ 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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/8de44ad586620f4e7790a5fe9e43e4ebee68b9be

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Five lectures on optimal transportation: Geometry, regularity and applications (2010, unknown): “The existence of −(u, v) ∈ Lipc such that γ vanishes outside the zero set of the non-negative function k(x, y) = c(x, y) + u(x) + v(y) ≥ 0 on M+ ×M− is necessary and sufficient for the optimality of γ ∈ Γ(µ+, ν−) with respect to c ∈ C(M+ ×M−).”
Hölder Continuity and Injectivity of Optimal Maps (2011, background): “We also assume that there exists a ball B such that Q ⊂ B ⊂ 4B ⊂ Uȳ, and that diam(Q) ≤ ε′c.”