Criteria for convexity in Banach spaces

Kanellopoulos, Vassilis

doi:10.1090/S0002-9939-00-05300-4

Criteria for convexity in Banach spaces
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Proc. Amer. Math. Soc. 128 (2000), 2725-2733 Request permission

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

G.Aumann On a topological characterization of compact convex point sets Ann. of Math 37, (1936) pp. 443-447.
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, DOI 10.1007/BFb0069321
James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
I. Fàry A characterization of convex bodies Am. Math. Monthly, vol 69, (1962) pp. 25-31.
P. Habala-P. Hajek-V. Zizler Introduction to Banach spaces Matfyz. Press 1996.
A. Kosiński, Note on star-shaped sets, Proc. Amer. Math. Soc. 13 (1962), 931–933. MR 143104, DOI 10.1090/S0002-9939-1962-0143104-X
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282