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]**Errett Bishop and R. R. Phelps,*The support functionals of a convex set*, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc., Providence, R.I., 1963, pp. 27–35. MR**0154092****[2]**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****[3]**Joseph Diestel,*Geometry of Banach spaces—selected topics*, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR**0461094****[4]**J. Diestel and J. J. Uhl Jr.,*Vector measures*, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical Surveys, No. 15. MR**0453964****[5]**Pando G. Georgiev,*Almost all convex, closed and bounded subsets of a Banach space are dentable*, Mathematics and mathematical education (Sunny Beach (Sl″nchev Bryag), 1985) B″lgar. Akad. Nauk, Sofia, 1985, pp. 355–361 (English, with Bulgarian summary). MR**805373****[6]**John R. Giles,*Convex analysis with application in the differentiation of convex functions*, Research Notes in Mathematics, vol. 58, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR**650456****[7]**J. R. Giles, D. A. Gregory, and Brailey Sims,*Characterisation of normed linear spaces with Mazur’s intersection property*, Bull. Austral. Math. Soc.**18**(1978), no. 1, 105–123. MR**0493266**, https://doi.org/10.1017/S0004972700007863**[8]**Gilles Godefroy,*Points de Namioka. Espaces normants. Applications à la théorie isométrique de la dualité*, Israel J. Math.**38**(1981), no. 3, 209–220 (French, with English summary). MR**605379**, https://doi.org/10.1007/BF02760806**[9]**G. Godefroy, S. Troyanski, J. Whitefield, and V. Zizler,*Locally uniformly rotund renorming and injections into 𝑐₀(Γ)*, Canad. Math. Bull.**27**(1984), no. 4, 494–500. MR**763053**, https://doi.org/10.4153/CMB-1984-079-8**[10]**K. Kuratovskiĭ,*\cyr Topologiya. Tom I*, Translated from the English by M. Ja. Antonovskiĭ. With a preface by P. S. Aleksandrov, Izdat. “Mir”, Moscow, 1966 (Russian). MR**0217750****[11]**S. Mazur,*Über schwach Konvengenz in den Raumen*), Studia Math.**4**(1933), 128-133.**[12]**R. R. Phelps,*A representation theorem for bounded convex sets*, Proc. Amer. Math. Soc.**11**(1960), 976–983. MR**0123172**, https://doi.org/10.1090/S0002-9939-1960-0123172-X**[13]**Francis Sullivan,*Dentability, smoothability and stronger properties in Banach spaces*, Indiana Univ. Math. J.**26**(1977), no. 3, 545–553. MR**0438088**, https://doi.org/10.1512/iumj.1977.26.26042

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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