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

Authors: Nguyen To Nhu and Le Hoang Tri
Journal: Proc. Amer. Math. Soc. 120 (1994), 1261-1265
MSC: Primary 54C55; Secondary 54D45
MathSciNet review: 1152989
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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).

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  • [BP] C. Bessaga and A. Pelczynski, Selected topics in infinite dimensional topology, PWN, Warszawa, 1975. MR 0478168 (57:17657)
  • [DG] J. Dugundji and A. Granas, Fixed point theory. I, PWN, Warszawa, 1982. MR 660439 (83j:54038)
  • [DT] T. Dobrowolski and H. Torunczyk, Separable complete $ ANR$'s admitting a group structure are Hilbert manifolds, Topology Appl. 12 (1981), 229-235. MR 623731 (83a:58007)
  • [G] R. Geoghegan, Open problems in infinite dimensional topology, Topology Proc. 4 (1979), 287-333. MR 583711 (82a:57015)
  • [K] V. L. Klee, Shrinkable neighbourhoods in Hausdorff linear spaces, Math. Ann. 141 (1960), 281-285. MR 0131149 (24:A1003)
  • [KM] M. G. Krein and D. P. Milman, On extreme points of regular convex sets, Studia Math. 9 (1940), 133-138. MR 0004990 (3:90a)
  • [KP] N. J. Kalton and N. T. Peck, A re-examination of Roberts' example of compact convex sets with no extreme points, Math. Ann. 253 (1980), 89-101. MR 597819 (82h:46055)
  • [N1] Nguyen To Nhu, Investigating the $ ANR$-property of metric spaces, Fund. Math. 124 (1984), 243-254. MR 774515 (86d:54018)
  • [N2] -, Remarks on measurable function spaces, Acta Math. Vietnam 12 (1987), 85-92. MR 919632 (89b:46039)
  • [N3] -, The group of measure preserving transformations of the unit interval is an absolute retract, Proc. Amer. Math. Soc. 110 (1990), 515-522. MR 1009997 (90m:28024)
  • [R1] J. W. Roberts, A compact convex set with no extreme points, Studia Math. 60 (1977), 255-266. MR 0470851 (57:10595)
  • [R2] -, Pathological compact convex sets in the spaces $ {L_p},\;0 \leqslant p < 1$, The Altgeld Book, Univ. of Illinois, 1976.
  • [Re] S. Rolewicz, Metric linear spaces, PWN, Warszawa, 1972. MR 0438074 (55:10993)
  • [Ru] W. Rudin, Functional analysis, McGraw-Hill, New York, 1973. MR 0365062 (51:1315)

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Keywords: Needle point, needle point space, extreme point, convex set, AR, fixed point property, Hilbert cube, admissible convex set
Article copyright: © Copyright 1994 American Mathematical Society

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