Proceedings of the American Mathematical Society

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

Author(s): Petr Holicky; Miklós Laczkovich
Journal: Proc. Amer. Math. Soc. 132 (2004), 3345-3347.
MSC (2000): Primary 52A15, 28A05
Posted: April 21, 2004
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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.

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 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

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 52A15, 28A05

Petr Holicky
Affiliation: Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 95 Prague 8, Czech Republic
Email: holicky@karlin.mff.cuni.cz

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
Email: laczk@cs.elte.hu

DOI: 10.1090/S0002-9939-04-07445-3
PII: S 0002-9939(04)07445-3
Keywords: Exposed points, convex compact sets
Received by editor(s): February 5, 2003
Received by editor(s) in revised form: July 21, 2003
Posted: 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 GACR 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.
Copyright of article: Copyright 2004, American Mathematical Society

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