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Descriptive properties of the set of exposed points of compact convex sets in $\mathbb{R} ^3$

Authors: Petr Holicky and Miklós Laczkovich
Journal: Proc. Amer. Math. Soc. 132 (2004), 3345-3347
MSC (2000): Primary 52A15, 28A05
Published electronically: April 21, 2004
MathSciNet review: 2073311
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Abstract | References | Similar Articles | Additional Information

Abstract: We construct a compact convex subset of $\mathbb R^3$ such that the set of its exposed points is not the intersection of an $F_{\sigma}$ set and a $G_{\delta}$ set. The existence of such a set answers a question posed by G. Choquet, H.H. Corson and V.L. Klee.

References [Enhancements On Off] (What's this?)

  • 1. G. Choquet, H. Corson and V. Klee, Exposed points of convex sets, Pacific J. Math. 17 (1966), 33-43. MR 33:6335
  • 2. H.H. Corson, A compact convex set in $E^3$ whose exposed points are of the first category, Proc. Amer. Math. Soc. 16 (1965), 1015-1021. MR 31:5147
  • 3. V.L. Klee, Extremal structure of convex sets, II, Pacific J. Math. 69 (1958), 90-104. MR 19:1065b
  • 4. R.T. Rockafellar, Convex Analysis, Princeton, New Jersey, Princeton University Press 1970. MR 43:445

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Additional Information

Petr Holicky
Affiliation: Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 95 Prague 8, Czech Republic

Miklós Laczkovich
Affiliation: Department of Analysis, Eötvös Loránd University, Budapest, Pázmány Péter sétány 1/c, Hungary – and – Department of Mathematics, University College London, Gower Street, London WC1E 6BT, England

Keywords: Exposed points, convex compact sets
Received by editor(s): February 5, 2003
Received by editor(s) in revised form: July 21, 2003
Published electronically: April 21, 2004
Additional Notes: The first author was supported by the “Mathematics in Information Society” project carried out by Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, in the framework of the European Community’s “Confirming the International Role of Community Research” program. The research was partly supported also by grants GAČR 201/03/0931 and MSM 113200007
The second author was partially supported by the Hungarian National Foundation for Scientific Research Grant No. T032042
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2004 American Mathematical Society