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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
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.

References [Enhancements On Off] (What's this?)

  • [1] G. Choquet, Lectures on analysis, Benjamin, New York, 1969.
  • [2] G. A. Edgar, Extremal integral representations, J. Functional Analysis 23 (1976), no. 2, 145–161. MR 0435797
  • [3] Richard Haydon, Some more characterizations of Banach spaces containing 𝑙₁, Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 2, 269–276. MR 0423047
  • [4] Robert R. Phelps, Lectures on Choquet’s theorem, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0193470

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Article copyright: © Copyright 1983 American Mathematical Society