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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G. Choquet, Lectures on analysis, Benjamin, New York, 1969.
- G. A. Edgar, Extremal integral representations, J. Functional Analysis 23 (1976), no. 2, 145–161. MR 0435797, DOI https://doi.org/10.1016/0022-1236%2876%2990072-0
- Richard Haydon, Some more characterizations of Banach spaces containing $l_{1}$, Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 2, 269–276. MR 423047, DOI https://doi.org/10.1017/S0305004100052890
- 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
