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Criteria for convexity in Banach spaces
Author(s):
Vassilis
Kanellopoulos
Journal:
Proc. Amer. Math. Soc.
128
(2000),
2725-2733.
MSC (2000):
Primary 52A07, 46B20;
Secondary 46B22
Posted:
February 28, 2000
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Abstract:
In this paper two convexity criteria are proven. The first one characterizes compact convex sets in a locally convex space and extends a previous result by G. Aumann, while the second one characterizes closed bounded convex sets with the Radon-Nikodým property in a Banach space.
References:
- [A]
- G.Aumann On a topological characterization of compact convex point sets Ann. of Math 37, (1936) pp. 443-447.
- [B]
- R. Bourgin Geometric aspects of convex sets with the Radon-Nikodým property Lecture notes in Math. Springer-Verlag 1983. MR 85d:46023
- [Du1]
- J. Dugundji Topology Allyn and Bacon Boston 1966. MR 33:1824
- [Du2]
- J. Dugundji An extension of Tietze's Theorem. Pac. J. Math.1 (1951) pp. 353-367. MR 13:373c
- [F]
- I. Fàry A characterization of convex bodies Am. Math. Monthly, vol 69, (1962) pp. 25-31.
- [HHZ]
- P. Habala-P. Hajek-V. Zizler Introduction to Banach spaces Matfyz. Press 1996.
- [K]
- A. Kosinski Note on star shaped sets Proc. Amer. Math. Soc. 13, (1962) 931-933. MR 26:669
- [S]
- R. Schneider Convex bodies. The Brun-Minkowski Theory. Cambridge University Press 1993. MR 94d:52007
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Additional Information:
Vassilis
Kanellopoulos
Affiliation:
Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece
Email:
bkanel@math.uoa.gr
DOI:
10.1090/S0002-9939-00-05300-4
PII:
S 0002-9939(00)05300-4
Keywords:
Convex set,
locally convex space,
Banach space,
extreme point,
slice,
contractible set,
Radon-Nikod\'{y}m property
Received by editor(s):
October 6, 1997
Received by editor(s) in revised form:
October 6, 1998
Posted:
February 28, 2000
Additional Notes:
This research was financially supported by I.K.Y. (National Foundation for Scholarships)
Communicated by:
Dale E. Alspach
Copyright of article:
Copyright
2000,
American Mathematical Society
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