Spheres in infinite-dimensional normed spaces are Lipschitz contractible

Abstract

Let X X be an infinite-dimensional normed space. We prove the following: (i) The unit sphere { x ∈ X : ‖ x ‖ = 1 } \{ x \in X:\left \| x \right \| = 1\} is Lipschitz contractible. (ii) There is a Lipschitz retraction from the unit ball of X X onto the unit sphere. (iii) There is a Lipschitz map T T of the unit ball into itself without an approximate fixed point, i.e. inf { ‖ x − T x ‖ : ‖ x ‖ ⩽ 1 } > 0 \inf \{ \left \| {x - Tx} \right \|:\left \| x \right \| \leqslant 1\} > 0 .