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On the geometric structure of convex sets with the RNP

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Vassilis Kanellopoulos
Vassilis Kanellopoulos
Affiliation:
Department of Mathematics, National Technical University of Athens, Zagrafou Campus, 15780, Athens, Greece. E-mail: bkanel@math.ntua.gr
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Abstract

An extension of Asplund's theorem concerning the n-extreme and the n-exposed points of a convex body in ℝn and an extension of Liberman's characterization of convexity are given for closed convex bounded sets with the RNP.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 50 , Issue 1-2 , December 2003 , pp. 73 - 85
Copyright
Copyright © University College London 2003

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References

[As]Asplund, E.. A k:-extreme point is the limit of k-exposed points. Israel J. Math. 1 (1963), 161–162.CrossRefGoogle Scholar
[A]Aumann, G.. On a topological characterization of compact convex point sets. Ann. Math 37 (1936), 443–447.CrossRefGoogle Scholar
[B1]Bourgin, D. G.. A condition for finite dimensional convexity. Proc. Amer. Math. Soc. 16. (1965), 1311–1312.CrossRefGoogle Scholar
[B]Bourgin, R.. Geometric aspects of convex sets with the Radon-Nikodým property. Lecture Notes in Math., Springer-Verlag (1983).Google Scholar
[Du1]Dugundji, J.. Topology. Allyn and Bacon (Boston, 1966).Google Scholar
[Du2]Dugundji, J.. An extension of Tietze's Theorem. Pacific J. Math. 1 (1951), 353–367.CrossRefGoogle Scholar
[K]Kanellopoulos, V.. Criteria for convexity in Banach spaces. Proc. Amer. Math. Soc. 128 (2000), 2725–2733.CrossRefGoogle Scholar
[L]Liberman, I.. On certain characteristic properties of convex bodies. Mat. Sb. N. S. 13 (1943), 236–262. (In Russian, see also Math. Rev. 6. p. 184.)Google Scholar
[LLT]Bor-Liu-Lin, , Pei-Kee-Lin, and Troyanski, S. L.. A characterization of denting points of a closed bounded convex set. Longhorn Notes, U. T. Functional Analysis Seminar, 1985–1986. The University of Texas at Austin.Google Scholar
[ML]Mani-Levitska, P.. Characterizations of convex sets. Handbook of Convex Geometry, North Holland (1993), 19–41.CrossRefGoogle Scholar
[HHZ]Habala, P.–Hajek, P.–Zizler, V.. Introduction to Banach Spaces. Matfyz. Press (1996).Google Scholar
[S]Schneider, R.. Convex Bodies: the Brunn-Minkowski Theory. Cambridge University Press (1993).CrossRefGoogle Scholar
[Str]Straszewicz, S.. Über exponierte Punkte abgeschlossener Punktmengen. Fund. Math. 24 (1935), 139–143.CrossRefGoogle Scholar