Porosity phenomena of non-expansive, Banach space mappings

Abstract

For any non-trivial convex and bounded subset C of a Banach space, we show that outside of a σ-porous subset of the space of non-expansive mappings C → C, all mappings have the maximal Lipschitz constant one witnessed locally at typical points of C. This extends a result of Bargetz and the author from separable Banach spaces to all Banach spaces and the proof given is completely independent. We further establish a fine relationship between the classes of exceptional sets involved in this statement, captured by the hierarchy of notions of φ-porosity.

Generic properties of nonexpansive mappings on unbounded domains

“…The following lemma can be proved completely analogously to the corresponding one in [11]. Since the proof is quite short, we include it nevertheless to keep this article self-contained.…”

“…The aim of this section is to extend one of the results of [11] to the setting of nonexpansive mappings on unbounded convex subsets of hyperbolic spaces. More precisely, we want to prove the following theorem.…”

“…The proofs given in this section are based on modifications of ideas from [11]. The first step is to construct a mapping which is an isometry on a suitable set and close to a given nonexpansive mapping.…”

Typical Lipschitz mappings are typically non-differentiable

“…Proof. The approach we take to modify the mapping f to arrive at g is similar to that taken in [7,Lemma 3.3]. The conclusion of this lemma is valid for f if and only if it is valid for any mapping of the form f + p, where p : Q → Y is a constant mapping.…”

“…The next lemma is a generalisation of [7,Lemma 3.1] for normed spaces instead of convex sets. Lemma 3.1.…”