On absolute strong exposure for Lipschitz maps

Abstract

We introduce strongly exposing Lipschitz maps, a vector-valued extension of Weaver's peaking functions and a nonlinear analogue of absolutely strongly exposing operators. Our main result shows that a Lipschitz map is strongly exposing if and only if its canonical linearization is absolutely strongly exposing. This equivalence serves as a bridge between the linear and Lipschitz settings and enables us to transfer several results from the former to the latter. As applications, we establish norm-denseness and residuality results for strongly exposing Lipschitz maps, obtain an isomorphic characterization related to the denseness of strongly norm-attaining Lipschitz maps. We also investigate weak sequential denseness of strongly exposing Lipschitz maps. In particular, we prove that this property holds whenever the derived set of the underlying metric space is finite, while further examples show that, unlike for strongly norm-attaining Lipschitz maps, weak sequential denseness may fail beyond trivial cases.