Geometrical and topological properties of starlike bodies in Banach spaces
by
Daniel Azagra
Universidad Complutense de Madrid
Coauthors: Manuel Cepedello, Robert Deville, Tadeusz Dobrowolski, Mar Jimenez-Sevilla

Starlike bodies are objects of interest in nonlinear functional analysis because they are strongly related to n-homogeneous polynomials and smooth bump functions, and their geometrical and topological properties are thus worth studying. We establish several new results in this direction. First, we give a topological classification of such bodies in (separable) Banach spaces. Second, we prove that the boundary of every smooth Lipschitz bounded starlike body in an infinite dimensional Banach space is smoothly Lipschitz contractible, and such boundary is a smooth Lipschitz retract of the corresponding body (this is also the same as saying that Brouwer's fixed point theorem fails in infinite dimensions even for smooth Lipschitz self maps of balls or bounded starlike bodies). Third, we study the topological size of the set of tangent hyperplanes to a smooth starlike body in a Banach space and we relate this matter to the failure of James' theorem on the caracterization of reflexivity for the class of starlike bodies, and to the topological size of the range of the derivative of a smooth function between Banach spaces. We stress the interplay between infinite dimensional smooth topology and nonlinear functional analysis by relating questions about topological and geometrical properties of starlike bodies to other interesting problems on nonlinear analysis, such as the failure of Rolle's theorem in infinite dimensions and other ways of characterizing the smoothness properties of a Banach space.

Date received: March 27, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagw-06.