A Note on Minimal Displacement and Retraction Problems

Abstract

Ž 5 5. Let X, ? be an infinitely dimensional Banach space with the unit ball w x w x B and the unit sphere S. Since the works of Kakutani 5 and Klee 6 it is known that the Brouwer’s fixed point theorem does not hold in this setting. It means that there are continuous mappings T : B a B such that Tx / x for all x g B. Equivalently it means also that S is a retract of B thus there Ž . exists a continuous mapping retraction R : B a S such that Rx s x for all x g S. Much stronger results concerning the failure of Brouwers w x theorem for infinitely dimensional balls have been obtained by Nowak 10 , w x w x Benyamini and Sternfeld 1 , and Sternfeld and Lin 9 . Ž . Let, for any k G 0, L k denote the class of Lipschitz mappings with constant k. The following two statements are equivalent and true.