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

Resumen

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(T,Y) is not dense in Lip0(T,Y) for any Banach space Y, where 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 Lip0(M,Y) regardless Y but which contain isometric copies of [0,1] and so the Lipschitz-free space 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,R) is dense in Lip0(M,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.