On strongly norm attaining Lipschitz maps

TL;DR

This paper investigates the properties of Lipschitz maps that strongly attain their norm, revealing their topological and linear characteristics across various metric spaces, and establishing conditions for their density and geometric properties.

Contribution

It extends previous results by characterizing when the set of strongly norm attaining Lipschitz maps is not dense and links linear properties to the density of these maps.

Findings

The set of strongly norm attaining Lipschitz maps is not dense in length spaces or certain subsets of real numbers.

All linear properties that ensure Lindenstrauss property A also guarantee the norm density of strongly norm attaining maps.

The set of strongly norm attaining Lipschitz maps is weakly sequentially dense in all Lipschitz functions.

Abstract

We study the set $\operatorname{SNA}(M,Y)$ of those Lipschitz maps from a (complete pointed) metric space $M$ to a Banach space $Y$ which (strongly) attain their Lipschitz norm (i.e.\ the supremum defining the Lipschitz norm is a maximum). Extending previous results, we prove that this set is not norm dense when $M$ is a length space (or local) or when $M$ is a closed subset of $\mathbb{R}$ with positive Lebesgue measure, providing new examples which have very different topological properties than the previously known ones. On the other hand, we study the linear properties which are sufficient to get Lindenstrauss property A for the Lipschitz-free space $\mathcal{F}(M)$ over $M$, and show that all of them actually provide the norm density of $\operatorname{SNA}(M,Y)$ in the space of all Lipschitz maps from $M$ to any Banach space $Y$. Next, we prove that $\operatorname{SNA}(M,\mathbb{R})$ is weakly sequentially dense in the space of all Lipschitz functions for all metric spaces $M$. Finally, we show that the norm of the bidual space of $\mathcal{F}(M)$ is octahedral provided the metric space $M$ is discrete but not uniformly discrete or $M'$ is infinite.