On strongly norm attaining Lipschitz maps

Abstract

We study the set $\mathrm{SNA}(M,Y)$ 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 $\mathrm{SNA}(M,Y)$ in the space of all Lipschitz maps from $M$ to any Banach space $Y$. Next, we prove that $\mathrm{SNA}(M,\mathbb{R})$ 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 $M^{\prime }$ is infinite.

28 pages, electronically published in the Journal of Functional Analysis

References in corpus (5)

• On the structure of Lipschitz-free spaces
• Remarks on diameter 2 properties
• Norm attaining Lipschitz functionals
• On the preserved extremal structure of Lipschitz-free spaces
• Complete metric spaces with property (Z) are length metric spaces

Cited by in corpus (11)

• Purely 1-unrectifiable metric spaces and locally flat Lipschitz functions
• Embeddings of Lipschitz-free spaces into ${\ell }_1$
• The Bishop--Phelps--Bollobás property for Lipschitz maps
• Integral representation and supports of functionals on Lipschitz spaces
• Points of differentiability of the norm in Lipschitz-free spaces
• Residuality in the set of norm attaining operators between Banach spaces
• Emerging notions of norm attainment for Lipschitz maps between Banach spaces
• Normal functionals on Lipschitz spaces are weak${}^{\ast }$ continuous
• Norm attaining Lipschitz maps toward vectors
• On a new space consisting of norm maintaining functions
• The Daugavet and Delta-constants of points in Banach spaces