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extracta mathematicae Vol. 17, Num. 2, 151 200 (2002) Geometrical and Topological Properties of Bumps and

extracta mathematicae Vol. 17, N´um. 2, 151 ­ 200 (2002)
Geometrical and Topological Properties of Bumps and
Starlike Bodies in Banach Spaces
Daniel Azagra, Mar Jim´enez-Sevilla
Departamento de An´alisis Matem´atico, Facultad de Ciencias Matem´aticas,
Universidad Complutense, 28040 Madrid, Spain
e-mail: Daniel Azagra@mat.ucm.es, mm jimenez@mat.ucm.es
(Survey paper presented by J. Jaramillo)
AMS Subject Class. (2000): 46B20, 46G05, 57N20 Received January 21, 2002
1. Introduction
A bump b on a Banach space X is a (most often smooth, at least continu-
ous) function with bounded nonempty support, supp(b) = {x X : b(x) = 0}.
The existence of smooth bump functions on a Banach space X is closely re-
lated in several ways to the (linear and nonlinear) structure of the space X,
and has often important consequences on its geometrical properties (see [27]).
In connection with bump functions there is the class of starlike bodies, which,
perhaps, have not yet received the attention that they are worth.
A closed subset A of a Banach space X is said to be a starlike body if
there exists a point x0 in the interior of A such that every ray emanating from


Source: Azagra Rueda, Daniel - Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid


Collections: Mathematics