Examples and applications of the density of strongly norm attaining Lipschitz maps

Abstract

We study the density of the set SNA$(M,Y)$ of those Lipschitz maps from a (complete pointed) metric space $M$ to a Banach space $Y$ which strongly attain their norm (i.e., the supremum defining the Lipschitz norm is actually a maximum). We present new and somehow counterintuitive examples, and we give some applications.

First, we show that SNA$(\mathbb{T},Y)$ is not dense in Lip${}_0(\mathbb{T},Y)$ for any Banach space $Y$, where $\mathbb{T}$ denotes the unit circle in the Euclidean plane. This provides the first example of a Gromov concave metric space (i.e., every molecule is a strongly exposed point of the unit ball of the Lipschitz-free space) for which the density does not hold.

Next, we construct metric spaces $M$ satisfying that SNA$(M,Y)$ is dense in Lip${}_0(M,Y)$ regardless $Y$ but which contain isometric copies of $[0,1]$ and so the Lipschitz-free space $\mathcal{F}(M)$ fails the Radon–Nikodym property, answering in the negative a question posed by Cascales et al. in 2019 and by Godefroy in 2015. Furthermore, an example of such $M$ can be produced failing all the previously known sufficient conditions for the density of strongly norm attaining Lipschitz maps.

Finally, among other applications, we show that if $M$ is a boundedly compact metric space for which SNA$(M,\mathbb{R})$ is dense in Lip${}_0(M,\mathbb{R})$, then the unit ball of the Lipschitz-free space on $M$ is the closed convex hull of its strongly exposed points. Further, we prove that given a compact metric space $M$ which does not contain any isometric copy of $[0,1]$ and a Banach space $Y$, if SNA$(M,Y)$ is dense, then SNA$(M,Y)$ actually contains an open dense subset.