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.

**[1]**E. Bishop and R. R. Phelps,*The support functionals of a convex set*, Convexity, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc., Providence, R. I., 1963, pp. 27-35. MR**0154092 (27:4051)****[2]**R. D. Bourgin,*Geometric aspects of convex sets with the Radon-Nikodym property*, Lecture Notes in Math., vol. 993, Amer. Math. Soc., Providence, R. I., 1983. MR**704815 (85d:46023)****[3]**J. Diestel,*Geometry of Banach spaces--selected topics*, Lecture Notes in Math., vol. 485, Springer-Verlag, Berlin and New York, 1975. MR**0461094 (57:1079)****[4]**J. Diestel and J. Uhl,*Vector measures*, Math. Surveys, vol. 15, Amer. Math. So.c, Providence, R.I.L, 1977. MR**0453964 (56:12216)****[5]**P. Gr. Georgiev,*Almost all closed, convex and bounded subsets of a Banach space are dentable*, Pro.c 14th Spring Conf. Union Bulg. Math., 1985, 355-361. MR**805373 (87h:46039)****[6]**J. R. Giles,*Convex analysis with application in differentiation of convex functions*, Pitman Advanced Publ. Prog., 1982. MR**650456 (83g:46001)****[7]**J. R. Giles, P. A. Gregory and B. Sims,*Characterization of normed linear spaces with Mazur's intersection property*, Bull. Austral. Math. Soc.**18**(1978), 105-123. MR**0493266 (58:12295)****[8]**G. Godefroy,*Points de Namióka, espaces normants, applications a la théorie isométrique de la dualité*, Israel J. Math.**38**(1981), 209-220. MR**605379 (82b:46014)****[9]**G. Godefroy, S. Troyanski, J. Whitfield and V. Zizler,*Locally uniformly rotund renorming and injections into*), Canad. Math. Bull.**27**(1984), 494-500. MR**763053 (86a:46016)****[10]**K. Kuratowski,*Topology*I, "Mir", Moscow, 1966. (Russian) MR**0217750 (36:839)****[11]**S. Mazur,*Über schwach Konvengenz in den Raumen*), Studia Math.**4**(1933), 128-133.**[12]**R. R. Phelps,*A representation of bounded convex sets*, Proc. Amer. Math. Soc.**11**(1960), 976-983. MR**0123172 (23:A501)****[13]**F. Sullivan,*Dentability, smoothability and stronger properties in Banach spaces*, Indiana Math. J.**26**(1977), 545-553. MR**0438088 (55:11007)**

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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