Mazur's intersection property and a KreĭnMilman type theorem for almost all closed, convex and bounded subsets of a Banach space
Author:
Pando Grigorov Georgiev
Journal:
Proc. Amer. Math. Soc. 104 (1988), 157164
MSC:
Primary 46B20
MathSciNet review:
958059
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Abstract: Let (resp. ) be the set of all closed, convex and bounded (resp. compact and convex) subsets of a Banach space (resp. of its dual ) furnished with the Hausdorff metric. It is shown that if there exists an equivalent norm in with dual such that has Mazur's intersection property and has Mazur's intersection property, then (1) there exists a dense subset of such that for every the strongly exposing functionals form a dense subset of ; (2) there exists a dense subset of such that for every the strongly exposing functionals form a dense subset of . In particular every is the closed convex hull of its strongly exposed points and every is the closed convex hull of its strongly exposed points.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198809580593
PII:
S 00029939(1988)09580593
Keywords:
Strongly exposed points,
strongly exposing functionals,
closed convex hull,
Mazur's intersection property,
sublinear functionals,
Fréchet differentiability
Article copyright:
© Copyright 1988
American Mathematical Society
