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On extremal nonexpansive mappings

Christian Bargetz, Michael Dymond, Katriin Pirk

TL;DR

The paper investigates extremality of nonexpansive self-mappings on bounded convex subsets of Banach spaces, focusing on surjective isometries. It develops a general reduction showing that extremality of surjective isometries follows from the extremality of the identity under broad hypotheses, including spaces with the Radon-Nikodym property and C(K) spaces, and provides detailed proofs for C(K) and c0. It further characterizes linear extremal maps on B_{c0} and demonstrates, using constructive perturbations, that surjective isometries are extremal in these settings. Finally, it establishes that, in the Baire category sense, typical nonexpansive mappings have only highly porous sets of convex decompositions, i.e., they are near-extremal in a precise, measure-zero sense.

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.

On extremal nonexpansive mappings

TL;DR

The paper investigates extremality of nonexpansive self-mappings on bounded convex subsets of Banach spaces, focusing on surjective isometries. It develops a general reduction showing that extremality of surjective isometries follows from the extremality of the identity under broad hypotheses, including spaces with the Radon-Nikodym property and C(K) spaces, and provides detailed proofs for C(K) and c0. It further characterizes linear extremal maps on B_{c0} and demonstrates, using constructive perturbations, that surjective isometries are extremal in these settings. Finally, it establishes that, in the Baire category sense, typical nonexpansive mappings have only highly porous sets of convex decompositions, i.e., they are near-extremal in a precise, measure-zero sense.

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 -spaces for compact Hausdorff . We also conclude that the typical, in the sense of Baire category, nonexpansive mapping is close to being extremal.
Paper Structure (5 sections, 25 theorems, 58 equations)

This paper contains 5 sections, 25 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General observations on the extremality of surjective isometries
  4. Extremality of surjective isometries on the unit ball of the space $C(K)$ or $c_0$
  5. On the extremality of the typical nonexpansive mapping

Key Result

Theorem 1.1

Let $X$ be a Banach space satisfying at least one of the following conditions: Then every surjective isometry $\mathbb{B}_{X}\to \mathbb{B}_{X}$ is extremal in the space of non-expansive mappings $\mathbb{B}_{X}\to \mathbb{B}_{X}$.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Lemma 3.3
  • ...and 45 more