Smooth Lipschitz retractions of starlike bodies onto their boundaries in infinite-dimensional Banach spaces

Abstract

Let X be an infinite-dimensional 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 C p Lipschitz contractible. (2) There is a C p Lipschitz retraction from A onto ∂A. (3) There is a C p Lipschitz map T : A −→ A with no approximate fixed points.