Mazur's intersection property and a Kreĭn-Milman 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), 157-164

MSC:
Primary 46B20

DOI:
https://doi.org/10.1090/S0002-9939-1988-0958059-3

MathSciNet review:
958059

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Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9939-1988-0958059-3

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