Plasticity of the unit ball of the real Banach space ${\ell }_{\infty }$

Abstract

We prove that the closed unit ball of the real Banach space ${\ell }_{\infty }$ is plastic, that is, every non-expansive bijection from the unit ball onto itself is an isometry. The main step is to show that every non-expansive bijection of this ball maps extreme points to extreme points. This is done by using elementary coverings of the unit ball by balls of radius one. The conclusion then follows from a theorem by Fakhoury. The same argument also shows that, for arbitrary $\Gamma$, every non-expansive bijection of $B_{{\ell }_{\infty }(\Gamma )}$ preserves extreme points.