Choquet simplexes whose set of extreme points is K-analytic
Annales de l'Institut Fourier, Volume 35 (1985) no. 3, p. 195-206

We construct a Choquet simplex K whose set of extreme points T is 𝒦-analytic, but is not a 𝒦-Borel set. The set T has the surprising property of being a K σδ set in its Stone-Cech compactification. It is hence an example of a K σδ set that is not absolute.

On construit un simplexe de Choquet dont l’ensemble des points extrémaux T est 𝒦-analytique, mais n’est pas 𝒦-Borélien. L’ensemble T est un K σδ dans sa compactification de Stone-Cech. C’est donc un exemple d’ensemble K σδ qui n’est pas absolu.

@article{AIF_1985__35_3_195_0,
     author = {Talagrand, Michel},
     title = {Choquet simplexes whose set of extreme points is $K$-analytic},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {35},
     number = {3},
     year = {1985},
     pages = {195-206},
     doi = {10.5802/aif.1024},
     zbl = {0564.46008},
     mrnumber = {87a:46022},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1985__35_3_195_0}
}
Talagrand, Michel. Choquet simplexes whose set of extreme points is $K$-analytic. Annales de l'Institut Fourier, Volume 35 (1985) no. 3, pp. 195-206. doi : 10.5802/aif.1024. http://www.numdam.org/item/AIF_1985__35_3_195_0/

[1] G. Choquet, Lectures on analysis, New York, W.A. Benjamin, 1969 (Math. Lecture Notes series). | Zbl 0181.39602

[2] Z. Frolick, A survey of separable descriptive theory of sets and spaces, Czechoslovak Math. J., 20 (1967), 406-467. | Zbl 0223.54028

[3] R. Haydon, An extreme point criterion for separability of a dual Banach space, and a new proof of a theorem of Corson, Quarterly J. Math., 27 (1976), 379-385. | MR 58 #12293 | Zbl 0335.46012

[4] B. Macgibbon, A criterion for the metrizability of a compact convex set in terms of the set of extreme points, J. Funct. Anal., 11 (1972), 385-392. | MR 49 #7723 | Zbl 0281.46001

[5] R. Phelps, Lectures on Choquet's theorem, Van Nostrand Math. studies, 7 (1966). | MR 33 #1690 | Zbl 0135.36203

[6] M. Talagrand, Géométrie du simplexe des moyennes, J. Funct. Anal., 33 (1919), 304-333. | Zbl 0431.43001

[7] M. Talagrand, Espaces de Banach faiblement K-analytiques, Ann. of Math., 110 (1979), 407-438. | MR 81a:46021 | Zbl 0393.46019

[8] M. Talagrand, Sur les convexes compacts dont l'ensemble des points extrémaux est K-analytique, Bull. Soc. Math. France, 107 (1979), 49-53. | Numdam | MR 80j:46023 | Zbl 0422.46007