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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Three convex sets


Author: Michel Talagrand
Journal: Proc. Amer. Math. Soc. 89 (1983), 601-607
MSC: Primary 46A55
DOI: https://doi.org/10.1090/S0002-9939-1983-0718981-5
MathSciNet review: 718981
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Abstract: We construct a Choquet simplex $ K$ such that there is a universally measurable affine function $ f$ on $ K$, which satisfies the barycentric calculus, and is zero on the set of extreme points, but is not identically zero. We also construct a closed convex bounded set of a Banach space without extreme points, but such that each point is the barycenter of a maximal measure. Finally, we construct a closed bounded set $ L$ of $ {l^1}({\mathbf{R}})$ and a maximal measure on $ L$ which is supported by a weak Baire set which contains no extreme points.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0718981-5
Article copyright: © Copyright 1983 American Mathematical Society