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Supports of Extremal Doubly Stochastic Measures

Published online by Cambridge University Press:  20 November 2018

Abbas Moameni
Abbas Moameni*
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, ON KIS 5B6 e-mail: momeni@math.carleton.ca
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Abstract

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A doubly stochastic measure on the unit square is a Borel probability measure whose horizontal and vertical marginals both coincide with the Lebesgue measure. The set of doubly stochasticmeasures is convex and compact so its extremal points are of particular interest. The problem number 111 of Birkhoò is to provide a necessary and suõcient condition on the support of a doubly stochastic measure to guarantee extremality. It was proved by Beneš and Štepán that an extremal doubly stochastic measure is concentrated on a set which admits an aperiodic decomposition. Hestir and Williams later found a necessary condition which is nearly sufficient by further refining the aperiodic structure of the support of extremal doubly stochastic measures. Our objective in this work is to provide a more practical necessary and nearly sufficient condition for a set to support an extremal doubly stochastic measure.

Keywords

49Q15optimal mass transportdoubly stochastic measuresextremalityuniqueness
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 2 , 01 June 2016 , pp. 381 - 391
Copyright
Copyright © Canadian Mathematical Society 2016

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