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.