The support of discrete extremal measures with given marginals.

Denny, J. L.

doi:10.1307/mmj/1029002309

Cited by 16 publications

(11 citation statements)

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“…The paper presents results that extend that by Lindenstrauss (1965), Letac (1966), Denny (1980 and Benes, Stepan (1987, 1990. …”

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confidence: 78%

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Simplicial Measures and Sets of Uniqueness in the Marginal Problem

Štěpán¹

1993

Statistics &Amp; Risk Modeling

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Necessary and sufficient conditions for a probability measure Ρ defined on a product Χ χ Y of Polish spaces to be extremal in the set of the measures with same marginals as Ρ are found. The paper presents results that extend that by Lindenstrauss (1965), Letac (1966), Denny (1980 and Benes, Stepan (1987, 1990. IntroductionLet X be a Polish space. We shall denote by M(X) (Λίι(Χ)) the set of bounded Borel signed (probability) measures defined on X. Having a pair of Polish spaces Χ, Y and a measure m £ M(X x Y) we denote by m x and m y the projections of τη to X and Y, respectively (the marginal measures). Putting MARG(m) = (m"m y ) we define a map M(X χ Υ) -» M{X) X M(Y). We shall say that Ρ £ M^X χ Y) is a simplicial measure if it is an extremal point in C(P) = {Q e M,{X χ Y): MARG{Q) = MARG(P)}. A Borel set D C X x Y will be called a set of marginal uniqueness or an MU-set if MARG: Mi(D) Μι(Χ) χ M X {Y) is an injection. Denote by 5, MU, MUC, the set of simplicial measures, the family of all MU-sets and the family of all compact MU-sets, respectively.The purpose of the present paper is to discuss topological and measure-theoretical properties of sets that support a simplicial measure. Up to now only the case of at most countable X and. Y has been solved in a satisfactory manner. A brief summary to this research (Letac (1966), Diego, Germani (1972), Denny (1980, Mukerjee (1985) reads as follows:1.1 Suppose that the spaces X and Y are at most countable. Then:

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“…The paper presents results that extend that by Lindenstrauss (1965), Letac (1966), Denny (1980 and Benes, Stepan (1987, 1990. …”

supporting

confidence: 78%

“…Y has been solved in a satisfactory manner. A brief summary to this research (Letac (1966), Diego, Germani (1972), Denny (1980, Mukerjee (1985) reads as follows:1.1 Suppose that the spaces X and Y are at most countable. Then:…”

mentioning

confidence: 99%

Simplicial Measures and Sets of Uniqueness in the Marginal Problem

Štěpán¹

1993

Statistics &Amp; Risk Modeling

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“…, a i k−1 j k a i k j 1 must vanish-see Fig. 2 or Denny [37], where the terminology aperiodic is used. Similarly, a set…”

Section: Extremal Doubly Stochastic Measuresmentioning

confidence: 99%

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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“…We know by optimality that {M j (i, k)} i,k can be taken to belong to Ext (M), the set of extremal points of M, which consists of the so-called acyclic matrices (see [15]). They are those matrices belonging to M such that the following property holds:…”

Section: Equivalence With Previous Modelsmentioning

confidence: 99%

A Benamou–Brenier Approach to Branched Transport

Brasco¹,

Buttazzo²,

Santambrogio³

2011

SIAM J. Math. Anal.

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The problem of branched transportation aims to describe the movement of masses when, due to concavity effects, they have the interest to travel together as much as possible, because the cost for a path of length ℓ covered by a mass m is proportional to m α ℓ with 0 < α < 1. The optimization of this criterion let branched structures appear and is suitable to applications like road systems, blood vessels, river networks. . . Several models have been employed in the literature to present this transport problem, and the present paper looks at a dynamical one, similar to the celebrated Benamou-Brenier formulation of Kantorovitch optimal transport. The movement is represented by a path ρ t of probabilities, connecting an initial state µ 0 to a final state µ 1 , satisfying the continuity equation ∂ t ρ + div x q = 0 together with a velocity field v (with q = ρv being the momentum). The transportation cost to be minimized is non-convex and finite on atomic measures:

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The support of discrete extremal measures with given marginals.

Cited by 16 publications

References 0 publications

Simplicial Measures and Sets of Uniqueness in the Marginal Problem

Simplicial Measures and Sets of Uniqueness in the Marginal Problem

Optimal transportation, topology and uniqueness

A Benamou–Brenier Approach to Branched Transport

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