Every needle point space contains a compact convex AR-set with no extreme points

Nhu, Nguyen To; Tri, Le Hoang

doi:10.1090/S0002-9939-1994-1152989-0

Every needle point space contains a compact convex AR-set with no extreme points
HTML articles powered by AMS MathViewer

by Nguyen To Nhu and Le Hoang Tri

Proc. Amer. Math. Soc. 120 (1994), 1261-1265

DOI: https://doi.org/10.1090/S0002-9939-1994-1152989-0

PDF | Request permission

Abstract:

Every needle point contains a compact convex AR-set without any extreme points. In particular the following spaces contain such compact convex sets: (i) the spaces ${L_p},\;0 \leqslant p < 1$; (ii) the linear metric space constructed by Roberts (Studia Math. 60 (1977), 255-266).

References

Czesław Bessaga and Aleksander Pełczyński, Selected topics in infinite-dimensional topology, Monografie Matematyczne, Tom 58. [Mathematical Monographs, Vol. 58], PWN—Polish Scientific Publishers, Warsaw, 1975. MR 0478168
James Dugundji and Andrzej Granas, Fixed point theory. I, Monografie Matematyczne [Mathematical Monographs], vol. 61, Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1982. MR 660439
T. Dobrowolski and H. Toruńczyk, Separable complete ANRs admitting a group structure are Hilbert manifolds, Topology Appl. 12 (1981), no. 3, 229–235. MR 623731, DOI 10.1016/0166-8641(81)90001-8
Open problems in infinite-dimensional topology, Topology Proc. 4 (1979), no. 1, 287–338 (1980). MR 583711
Victor Klee, Shrinkable neighborhoods in Hausdorff linear spaces, Math. Ann. 141 (1960), 281–285. MR 131149, DOI 10.1007/BF01360762
M. Krein and D. Milman, On extreme points of regular convex sets, Studia Math. 9 (1940), 133–138 (English, with Ukrainian summary). MR 4990, DOI 10.4064/sm-9-1-133-138
N. J. Kalton and N. T. Peck, A re-examination of the Roberts example of a compact convex set without extreme points, Math. Ann. 253 (1980), no. 2, 89–101. MR 597819, DOI 10.1007/BF01578905
Nguyen To Nhu, Investigating the ANR-property of metric spaces, Fund. Math. 124 (1984), no. 3, 243–254. MR 774515, DOI 10.4064/fm-124-3-243-254
Nguyen To Nhu, Remarks on measurable function spaces, Acta Math. Vietnam. 12 (1987), no. 1, 85–92. MR 919632
Nguyen To Nhu, The group of measure preserving transformations of the unit interval is an absolute retract, Proc. Amer. Math. Soc. 110 (1990), no. 2, 515–522. MR 1009997, DOI 10.1090/S0002-9939-1990-1009997-6
James W. Roberts, A compact convex set with no extreme points, Studia Math. 60 (1977), no. 3, 255–266. MR 470851, DOI 10.4064/sm-60-3-255-266

Pathological compact convex sets in the spaces

Stefan Rolewicz, Metric linear spaces, Monografie Matematyczne, Tom 56. [Mathematical Monographs, Vol. 56], PWN—Polish Scientific Publishers, Warsaw, 1972. MR 0438074
Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062