Porosity phenomena of non-expansive, Banach space mappings

TL;DR

This paper extends the understanding of the typical behavior of non-expansive mappings in Banach spaces, showing that most have maximal local Lipschitz constants, with the exceptional set being $\sigma$-porous.

Contribution

It generalizes previous results from separable to all Banach spaces and introduces a detailed hierarchy of porosity notions for exceptional sets.

Findings

Most non-expansive mappings have local Lipschitz constant one.

The set of exceptions is $\sigma$-porous in the space of mappings.

A hierarchy of $\phi$-porosity notions describes the exceptional sets.

Abstract

For any non-trivial convex and bounded subset $C$ of a Banach space, we show that outside of a $\sigma$-porous subset of the space of non-expansive mappings $C\to C$, 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 $\phi$-porosity.