Counter-examples to some conjectures about doubly stochastic measures

Losert, Viktor

doi:10.2140/pjm.1982.99.387

Cited by 23 publications

(13 citation statements)

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“…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]. The difficulty of the problem resides partly in the fact that any geometrical characterization of optimality must be invariant under arbitrary measure-preserving transformations applied independently to the horizontal (abscissa) and vertical (ordinate) variables.…”

Section: Extremal Doubly Stochastic Measuresmentioning

confidence: 99%

“…This definition is equivalent to saying that there is no y 0 satisfying (12), i.e., the set of marked points in Let us conclude by recalling an example of an extremal doubly stochastic measure which does not lie on the graph of a single map, drawn from work of Gangbo and McCann [55] and Ahmad [3] on optimal transportation, and developed in an economic context by Chiappori, McCann, and Nesheim [27]. Other examples may be found in the work of Seethoff and Shiflett [92], Losert [69], Hestir and Williams [58], Gangbo and McCann [54], Uckelmann [101], McCann [76], and Plakhov [85].…”

Section: Uniqueness Of Optimal Transportationmentioning

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: Extremal Doubly Stochastic Measuresmentioning

confidence: 99%

Section: Uniqueness Of Optimal Transportationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimal transportation, topology and uniqueness

2011

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“…Remark. That it may not always be possible to choose these graphs to be Borel measurable follows (among other things) from an example of V. Losert [14] (see [1,20]). However the above example is simpler.…”

Section: It Would Be Interesting To Construct Such An Example Withmentioning

confidence: 99%

Sets with doubleton sections, good sets and ergodic theory

Kłopotowski¹,

Nadkarni

Sarbadhikari

et al. 2002

Fund. Math.

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Abstract.A Borel subset of the unit square whose vertical and horizontal sections are two-point sets admits a natural group action. We exploit this to discuss some questions about Borel subsets of the unit square on which every function is a sum of functions of the coordinates. Connection with probability measures with prescribed marginals and some function algebra questions is discussed.Introduction. Given non-empty sets X and Y , a subset of X × Y is called a matching if it is the graph of a one-to-one map of X onto Y . The question as to which subsets S of X × Y contain a matching has been of combinatorial and set-theoretic interest [12]. By a theorem of D. König [11], if each section S x = {y ∈ Y : (x, y) ∈ S}, x ∈ X, and S y = {x ∈ X : (x, y) ∈ S}, y ∈ Y , consists of exactly n elements (n finite; for infinite n see

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“…J. Lindenstrauss [2] shows that extremal dsm's are singular with respect to the Lebesgue measure on / . On the other hand, V. Losert [3] gives an example of an extremal dsm whose support is the whole of / . In this note we would like to discuss a different notion of support of a dsm which seems to describe better the set where the mass is concentrated.…”

Section: Introductionmentioning

confidence: 99%