On extreme doubly stochastic measures.

Brown, J. R.; Shiflett, Ray C.

doi:10.1307/mmj/1029000472

Cited by 11 publications

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“…and if v is a doubly stochastic measure which is 388 V. LOSERT absolutely continuous with respect to μ, then v -μ. In [2] this conjecture was confirmed by Brown and Shifiett for a class of extreme doubly stochastic measures which is geometrically related to permutation matrices. In Theorem 1 of our paper we will give a functional analytic characterization of measures μ that satisfy a slightly stronger property as above (v need not be positive -the result of [2] holds also for this stronger property).…”

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confidence: 75%

“…We will define a measure μ successively on the subalgebras Σ n <g) Σ r n . To assure that the limit measure is extremal, we will use an idea similar to [2] Thm. 1 and [3] Thm.…”

Section: Ii) Assume Thatmentioning

confidence: 99%

“…that can be found in the literature are concentrated on the graphs (or inverse graphs) of functions, mostly even linear functions and there was some common belief that any double stochastic measure must in some sense be made up from graphs (cf. the beginning of §2 of [2]). A functional analytic characterization of the e.d.s.m.…”

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confidence: 96%

“…has been given by Douglas [4] and Lindenstrauss [7]. Several authors have tried to generalize properties of permutation matrices to these measures, see e.g., [1], [2], [10]. One of the aims of this paper is to present some new constructions of e.d.s.m.…”

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confidence: 99%

“…While the measures in the first construction are still concentrated on two graphs, it turns out that even in this case the geometric interrelations are much more complicated. In [2] the following conjecture (attributed to J. Feldman) was mentioned: if μ is an e.d.s.m. and if v is a doubly stochastic measure which is 388 V. LOSERT absolutely continuous with respect to μ, then v -μ.…”

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confidence: 99%

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Counter-examples to some conjectures about doubly stochastic measures

Losert¹

1982

Pacific J. Math.

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confidence: 75%

“…We will define a measure μ successively on the subalgebras Σ n <g) Σ r n . To assure that the limit measure is extremal, we will use an idea similar to [2] Thm. 1 and [3] Thm.…”

Section: Ii) Assume Thatmentioning

confidence: 99%

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confidence: 96%

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confidence: 99%

mentioning

confidence: 99%

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Counter-examples to some conjectures about doubly stochastic measures

Losert¹

1982

Pacific J. Math.

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A Birkhoff Theorem for Doubly Stochastic Matrices with Vector Entries

Clapp

Shiflett

1980

Stud Appl Math

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The concept of a doubly stochastic matrix whose entries come from a convex subset of the unit square is defined. It is proved that the only convex subsets of the unit square which contain (0,0) and (1,1) and allow an extension of Birkhoff's characterization of the extreme points of the set of doubly stochastic matrices are parallelograms. A sufficient condition is given for a matrix to be extreme when the convex subset is not a parallelogram. o. IntroductionAn n X n matrix whose entries are taken from the closed unit interval and whose rows and columns sum to one was called doubly stochastic by Feller [9]. The importance of these matrices to applications of matrix theory goes back at least to Schur's work in 1923 [15], and their study has produced an abundance of theoretical papers which in turn have generated fertile areas of research. Two such important works are Hardy, Littlewood, and Polya's paper [10] The set of n X n doubly stochastic matrices is the convex hull of the permutation matrices (those with I in each row and column).Then in problem Ill, he suggests this theorem be generalized to the infinite case. The work of J. B. Isbell [11] and D. G. Kendall [12] concerns that countable case; that of J. Lindenstrauss [13], J. V. Ryff [14], and Brown and Shiflett [3] concerns the continuous case. Investigations continue on questions of extreme points of convex subsets of this set of matrices. Of particular interest are the papers by R. Brualdi [4][5][6][7][8]. It is proposed here that a new and apparently overlooked type of doubly stochastic matrix be considered which may be of

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Extremal Solutions in the Marginal Problem

Beneš

Štěpán

1991

Advances in Probability Distributions With Given Marginals

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On extreme doubly stochastic measures.

Cited by 11 publications

References 0 publications

Counter-examples to some conjectures about doubly stochastic measures

Counter-examples to some conjectures about doubly stochastic measures

A Birkhoff Theorem for Doubly Stochastic Matrices with Vector Entries

Extremal Solutions in the Marginal Problem

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