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MAZUR INTERSECTION PROPERTIES AND DIFFERENTIABILITY OF CONVEX FUNCTIONS IN BANACH SPACES

Published online by Cambridge University Press:  01 April 2000

P. G. GEORGIEV ,
A. S. GRANERO ,
M. JIMÉNEZ SEVILLA  and
J. P. MORENO
P. G. GEORGIEV
Affiliation:
Faculty of Mathematics and Informatics, Sofia University, St. Kl. Ohridski, Sofia, Bulgaria; pandogg@fmi.uni-sofia.bg
A. S. GRANERO
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Madrid 28040, Spain; granero@eucmax.sim.ucm.es, marjim@sunam1.mat.ucm.es
M. JIMÉNEZ SEVILLA
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Madrid 28040, Spain; granero@eucmax.sim.ucm.es, marjim@sunam1.mat.ucm.es
J. P. MORENO
Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Madrid 28049, Spain; josepedro.moreno@uam.es
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Abstract

It is proved that the dual of a Banach space with the Mazur intersection property is almost weak* Asplund. Analogously, the predual of a dual space with the weak* Mazur intersection property is almost Asplund. Through the use of these arguments, it is found that, in particular, almost all (in the Baire sense) equivalent norms on [lscr ]1(Γ) and [lscr ]∞(Γ) are Fréchet differentiable on a dense Gδ subset. Necessary conditions for Mazur intersection properties in terms of convex sets satisfying a Krein–Milman type condition are also discussed. It is also shown that, if a Banach space has the Mazur intersection property, then every subspace of countable codimension can be equivalently renormed to satisfy this property.

Type
Research Article
Information
Journal of the London Mathematical Society , Volume 61 , Issue 2 , April 2000 , pp. 531 - 542
Copyright
The London Mathematical Society 2000

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