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

We study the density of 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 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 $\operatorname{SNA}(\mathbb T,Y)$ is not dense in ${\mathrm{Lip}}_0(\mathbb T,Y)$ 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 $\operatorname{SNA}(M,Y)$ is dense in ${\mathrm{Lip}}_0(M,Y)$ regardless $Y$ but which contains an isometric copy of $[0,1]$ and so the Lipschitz-free space $\mathcal F(M)$ fails the Radon--Nikodým property, answering in the negative a posed question. Furthermore, an example $M$ can be produced failing all the previously known sufficient conditions to get the density of strongly norm attaining Lipschitz maps. Finally, among other applications, we prove that given a compact metric $M$ which does not contains any isometric copy of $[0,1]$ and a Banach space $Y$, if $\operatorname{SNA}(M,Y)$ is dense, then $\operatorname{SNA}(M,Y)$ actually contains an open dense subset and $B_{\mathcal F(M)}=\overline{\mathrm{co}}(\operatorname{str-exp}(B_{\mathcal F(M)}))$. Further, we show that if $M$ is a boundedly compact metric space for which $\operatorname{SNA}(M,\mathbb R)$ is dense in ${\mathrm{Lip}}_0(M,\mathbb R)$, then the unit ball of the Lipschitz-free space on $M$ is the closed convex hull of its strongly exposed points.