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Search: MSC category 49Q15 ( Geometric measure and integration theory, integral and normal currents [See also 28A75, 32C30, 58A25, 58C35] )

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1. CMB 2016 (vol 59 pp. 381)

Moameni, Abbas
Supports of Extremal Doubly Stochastic Measures
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 stochastic measures is convex and compact so its extremal points are of particular interest. The problem number 111 of Birkhoff (Lattice Theory 1948) is to provide a necessary and sufficient condition on the support of a doubly stochastic measure to guarantee extremality. It was proved by Beneš and Štėpán 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:optimal mass transport, doubly stochastic measures, extremality, uniqueness

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