Three convex sets
Abstract: We construct a Choquet simplex such that there is a universally measurable affine function on , 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 of and a maximal measure on which is supported by a weak Baire set which contains no extreme points.
-  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
-  Richard Haydon, Some more characterizations of Banach spaces containing 𝑙₁, Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 2, 269–276. MR 0423047, 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
- G. Choquet, Lectures on analysis, Benjamin, New York, 1969.
- G. A. Edgar, Extremal integral representation, J. Funct. Anal. 23 (1976), 145-165. MR 0435797 (55:8753)
- R. Haydon, Some more characterizations of Banach spaces containing , Math. Proc. Cambridge Philos. Soc. 80 (1976), 269-276. MR 0423047 (54:11031)
- R. R. Phelps, Lecture on Choquet's theorem, Van Nostrand, Princeton, N. J., 1966. MR 0193470 (33:1690)
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