Resumen: It was proved recently that a Banach space fails the Mazur intersection property if and only if the family of all closed, convex and bounded subsets which are intersections of balls is uniformly very porous. This paper deals with the geometrical implications of this result. It is shown that every equivalent norm on the space can be associated in a natural way with a
constant of porosity, whose interplay with the geometry of the space is then investigated. Among other things, we prove that this constant is closely related to the set of ε-differentiability points of the space and the set of r-denting points of the dual. We also obtain estimates for this constant in several classical spaces.
Palabras clave: Uniformly very porous; Set of weak denting points; Differentiability points; Constant of porosity; Mazur intersection property; Equivalent norm
Departamento: Fac. de CC. Matemáticas - Depto. de Análisis Matemático
ISSN: 0019-2082
CDU: 514.7
Notas: Supported in part by DGICYT Grant BMF-2000-0609.The authors wish to thank the C.E.C.M., the Department of Mathematics and Statistics at Simon Fraser University, and especially J. Borwein, for their hospitality during the preparation of this paper.
Resumen: We prove that every Banach space containing a complemented copy of c0 has an antiproximinal body for a suitable norm. If, in addition, the space is separable, there is a pair of antiproximinal norms. In particular, in a separable polyhedral space X, the set of all (equivalent) norms on X having an isomorphic antiproximinal norm is dense. In contrast,
it is shown that there are no antiproximinal norms in Banach spaces with the Convex Point of Continuity Property (CPCP). Other questions related to the existence of antiproximinal bodies are also discussed.
Palabras clave: Convex-functions; Intersection-properties; Continuity property; Differentiability; Sets; Point
Departamento: Fac. de CC. Matemáticas - Depto. de Análisis Matemático
ISSN: 0021-9045
CDU: 519.6
Notas: This work was begun while the second and third authors were visiting the CECM at the Simon Fraser University. The second author is indebted to Gilles Godefroy for his support, valuable suggestions, and many stimulating conversations.