We study the density of the set SNA of those Lipschitz maps from a (complete pointed) metric space to a Banach space 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 is not dense in Lip for any Banach space , where 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 satisfying that SNA is dense in Lip regardless but which contain isometric copies of and so the Lipschitz-free space 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 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 is a boundedly compact metric space for which SNA is dense in Lip, then the unit ball of the Lipschitz-free space on is the closed convex hull of its strongly exposed points. Further, we prove that given a compact metric space which does not contain any isometric copy of and a Banach space , if SNA is dense, then SNA actually contains an open dense subset.
Cite this article
Rafael Chiclana, Luis C. García-Lirola, Miguel Martín, Abraham Rueda Zoca, Examples and applications of the density of strongly norm attaining Lipschitz maps. Rev. Mat. Iberoam. 37 (2021), no. 5, pp. 1917–1951DOI 10.4171/RMI/1253