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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Borel classes of sets of extreme and exposed points in $\mathbb{R} ^n$


Authors: Petr Holicky and Tamás Keleti
Journal: Proc. Amer. Math. Soc. 133 (2005), 1851-1859
MSC (2000): Primary 03E15, 28A05, 52A20, 52A15
Published electronically: January 21, 2005
MathSciNet review: 2120287
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Abstract: It is known that the sets of extreme and exposed points of a convex Borel subset of $\mathbb{R} ^n$ are Borel. We show that for $n\ge 4$ there exist convex $G_{\delta}$ subsets of $\mathbb{R} ^n$ such that the sets of their extreme and exposed points coincide and are of arbitrarily high Borel class. On the other hand, we show that the sets of extreme and of exposed points of a convex set $C\subset \mathbb{R} ^3$ of additive Borel class $\alpha$ are of ambiguous Borel class $\alpha+1$. For proving the latter-mentioned results we show that the union of the open and the union of the closed segments of $C\cap \partial C$ are of the additive Borel class $\alpha$ if $C\subset\mathbb{R} ^3$ is a convex set of additive Borel class $\alpha$.


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Additional Information

Petr Holicky
Affiliation: Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 00 Prague 8, Czech Republic
Email: holicky@karlin.mff.cuni.cz

Tamás Keleti
Affiliation: Department of Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, Budapest, 1117, Hungary
Email: elek@cs.elte.hu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07743-9
PII: S 0002-9939(05)07743-9
Keywords: Convex set, extreme point, exposed point, Borel class
Received by editor(s): February 10, 2003
Received by editor(s) in revised form: August 23, 2003, and February 29, 2004
Published electronically: January 21, 2005
Additional Notes: The first author was supported by the “Mathematics in Information Society” project carried out by the 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 supported by OTKA grants F 029768 and F 043620
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2005 American Mathematical Society