Borel classes of sets of extreme and exposed points in

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 | References | Similar Articles | Additional Information

Abstract: It is known that the sets of extreme and exposed points of a convex Borel subset of are Borel. We show that for there exist convex subsets of 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 of additive Borel class are of ambiguous Borel class . For proving the latter-mentioned results we show that the union of the open and the union of the closed segments of are of the additive Borel class if is a convex set of additive Borel class .

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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

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