Supports of doubly stochastic measures

Hestir, Kevin; Williams, Stanley C.

doi:10.3150/bj/1193667816

Cited by 15 publications

(53 citation statements)

References 9 publications

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“…As in Hestir and Williams [58], the uniqueness theorem above implies extremality as an immediate consequence. Proof Suppose a measure γ ∈ (μ, ν) vanishes outside a numbered limb system S satisfying the hypotheses of the corollary.…”

Section: Numbered Limb Systems and Extremalitymentioning

confidence: 84%

“…It improves on Lemma 2.4 of [55] and various other antecedents, by using an argument from Villani's Theorem 5.28 [104] to extract μ-measurability of f as a conclusion rather that a hypothesis. As work of, e.g., Hestir and Williams [58] implies, although measures on graphs are extremal in (μ, ν), the converse is far from being true; this peculiarity is an inevitable consequence of the infinite divisibility of (X, μ). …”

Section: Measures On Graphs Are Push-forwardsmentioning

confidence: 97%

“…Significant further progress was made by Beneš and Štěpán, who showed every extremal doubly stochastic measure vanishes outside some acyclic subset S ⊂ X × Y [8]. Hestir and Williams [58] refined this condition, showing that it becomes sufficient under an additional Borel measurability hypothesis which, unfortunately, is not always satisfied. Some of the subtleties of the problem were indicated already by Losert's counterexamples [69].…”

Section: Extremal Doubly Stochastic Measuresmentioning

confidence: 99%

“…In this section we adapt Hestir and Williams [58] notion of a numbered limb systemalso called an axial forest or a limb numbering system-to X × Y . Using the axiom of choice, Hestir and Williams deduced from the acyclicity condition of Beneš and Štěpán [8] that each extremal doubly stochastic measure vanishes outside some numbered limb system.…”

Section: Numbered Limb Systems and Extremalitymentioning

confidence: 99%

“…The proof that the subtwist condition is sufficient for uniqueness relies on progress in Birkhoff's problem of characterizing extremal doubly stochastic measures on the square. This problem is esoteric and subtle: although still not completely resolved, substantial results have been obtained in the six decades since it was posed [8,39,58,67,69]. Highlights are surveyed below.…”

Section: Introductionmentioning

confidence: 99%

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Optimal transportation, topology and uniqueness

2011

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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 Communicated by N. Trudinger. Formerly titled Extremal doubly stochastic measures and optimal transportation.It is a pleasure to thank Nassif Ghoussoub and Herbert Kellerer, who provided early encouragement in this direction, and Pierre-Andre Chiappori, Ivar Ekeland, and Lars Nesheim, whose interest in economic applications fortified our resolve to persist. We thank Wilfrid Gangbo, Jonathan Korman, and Robert Pego for fruitful discussions, Nathan Killoran for useful references, and programs of the Banff International Research Station (2003) 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.

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Section: Numbered Limb Systems and Extremalitymentioning

confidence: 84%