On extremal nonexpansive mappings

TL;DR

This paper investigates the extremal properties of nonexpansive mappings in Banach spaces, showing that surjective isometries are extremal in many cases and that typical nonexpansive mappings are nearly extremal.

Contribution

It establishes conditions under which surjective isometries are extremal and analyzes the generic behavior of nonexpansive mappings in Banach spaces.

Findings

Surjective isometries are extremal in Banach spaces with the Radon-Nikodym property and $C(K)$-spaces.

Typical nonexpansive mappings are close to extremal in the Baire category sense.

The study extends understanding of extremal nonexpansive mappings in various Banach spaces.

Abstract

We study the extremality of nonexpansive mappings on a nonempty bounded closed and convex subset of a normed space (therein specific Banach spaces). We show that surjective isometries are extremal in this sense for many Banach spaces, including Banach spaces with the Radon-Nikodym property and all $C(K)$-spaces for compact Hausdorff $K$. We also conclude that the typical, in the sense of Baire category, nonexpansive mapping is close to being extremal.