Criteria for convexity in Banach spaces

Author:
Vassilis Kanellopoulos

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2725-2733

MSC (2000):
Primary 52A07, 46B20; Secondary 46B22

Published electronically:
February 28, 2000

MathSciNet review:
1662253

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Abstract | References | Similar Articles | Additional Information

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.

**[A]**G.Aumann*On a topological characterization of compact convex point sets*Ann. of Math 37, (1936) pp. 443-447.**[B]**Richard D. Bourgin,*Geometric aspects of convex sets with the Radon-Nikodým property*, Lecture Notes in Mathematics, vol. 993, Springer-Verlag, Berlin, 1983. MR**704815****[Du1]**James Dugundji,*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606****[Du2]**J. Dugundji,*An extension of Tietze’s theorem*, Pacific J. Math.**1**(1951), 353–367. MR**0044116****[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. Kosiński,*Note on star-shaped sets*, Proc. Amer. Math. Soc.**13**(1962), 931–933. MR**0143104**, 10.1090/S0002-9939-1962-0143104-X**[S]**Rolf Schneider,*Convex bodies: the Brunn-Minkowski theory*, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR**1216521**

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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:
http://dx.doi.org/10.1090/S0002-9939-00-05300-4

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

Published electronically:
February 28, 2000

Additional Notes:
This research was financially supported by I.K.Y. (National Foundation for Scholarships)

Communicated by:
Dale E. Alspach

Article copyright:
© Copyright 2000
American Mathematical Society