The extreme points of some convex sets in the theory of majorization |
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Affiliation: | Department of Economics, London School of Economics and Political Science, University of London, Houghton Street, London WC2A 2AE, United Kingdom |
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Abstract: | Let (A, %plane1D;49C;, μ) be a finite measure space, and let Ωµ, w+f denote the set of all nonnegative real-valued %plane1D;49C;-measurable functions on A weaklymajorized by a nonnegative function f, in the sense of Hardly, Littlewood and Pólya. For a nonatomic µ, the extreme points ofΩµ, w +f are shown to be the nonnegativefunctions obtained by taking a fraction (1−θ) of the largest values of and arranging them in any way on any subset of A of measure(1−θ), with values elsewhere set equal to zero. Topological properties of these extreme points are given. |
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