 
Summary: SMOOTH LIPSCHITZ RETRACTIONS OF STARLIKE BODIES
ONTO THEIR BOUNDARIES IN INFINITEDIMENSIONAL
BANACH SPACES
DANIEL AZAGRA AND MANUEL CEPEDELLO BOISO
Abstract. Let X be an infinitedimensional Banach space and let A be a Cp
Lipschitz bounded starlike body (for instance the unit ball of a smooth norm).
We prove that
(1) The boundary A is Cp
Lipschitz contractible.
(2) There is a Cp
Lipschitz retraction from A onto A.
(3) There is a Cp
Lipschitz map T : A  A with no approximate fixed points.
1. Introduction and main results
The well known Brouwer's fixed point theorem states that every continuous self
map of the unit ball of a finitedimensional Banach space admits a fixed point. This
is equivalent to saying that there is no continuous retraction from the unit ball onto
the unit sphere, or that the unit sphere is not contractible (the identity map on the
sphere is not homotopic to a constant map). This result is no longer true in infinite
dimensions (see [8]). In [14] B. Nowak showed that for several infinitedimensional
