Borel classes of sets of extreme and exposed points in $\mathbb {R}^n$
HTML articles powered by AMS MathViewer
- by Petr Holický and Tamás Keleti PDF
- Proc. Amer. Math. Soc. 133 (2005), 1851-1859 Request permission
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 $\alpha$.References
- Gustave Choquet, Harry Corson, and Victor Klee, Exposed points of convex sets, Pacific J. Math. 17 (1966), 33–43. MR 198176
- 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 180917, DOI 10.1090/S0002-9939-1965-0180917-5
- Petr Holický and Václav Komínek, Two remarks on the structure of sets of exposed and extreme points, Extracta Math. 15 (2000), no. 3, 547–561. MR 1826133
- P. Holický and M. Laczkovich, On the descriptive properties of the set of exposed points of a compact convex subset of $\mathbb {R}^3$, Proc. Amer. Math. Soc., to appear.
- J. E. Jayne and C. A. Rogers, The extremal structure of convex sets, J. Functional Analysis 26 (1977), no. 3, 251–288. MR 0511800, DOI 10.1016/0022-1236(77)90027-1
- Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
- K. Kuratowski, Topology. Vol. I, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe [Polish Scientific Publishers], Warsaw, 1966. New edition, revised and augmented; Translated from the French by J. Jaworowski. MR 0217751
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683
- C. A. Rogers, The convex generation of convex Borel sets in euclidean space, Pacific J. Math. 35 (1970), 773–782. MR 278189
Additional Information
- Petr Holický
- 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
- MR Author ID: 288479
- Email: elek@cs.elte.hu
- 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.
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 1851-1859
- MSC (2000): Primary 03E15, 28A05, 52A20, 52A15
- DOI: https://doi.org/10.1090/S0002-9939-05-07743-9
- MathSciNet review: 2120287