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

  • Rafael Chiclana

    Universidad de Granada, Spain
  • Luis C. García-Lirola

    Universidad de Zaragoza, Spain
  • Miguel Martín

    Universidad de Granada, Spain
  • Abraham Rueda Zoca

    Universidad de Murcia, Spain
Examples and applications of the density of strongly norm attaining Lipschitz maps cover
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Abstract

We study the density of the set SNA(M,Y)(M,Y) of those Lipschitz maps from a (complete pointed) metric space MM to a Banach space YY 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)(\mathbb T,Y) is not dense in Lip0(T,Y)_0(\mathbb T,Y) for any Banach space YY, where T\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 MM satisfying that SNA(M,Y)(M,Y) is dense in Lip0(M,Y)_0(M,Y) regardless YY but which contain isometric copies of [0,1][0,1] and so the Lipschitz-free space F(M)\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 MM 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 MM is a boundedly compact metric space for which SNA(M,R)(M,\mathbb R) is dense in Lip0(M,R)_0(M,\mathbb R), then the unit ball of the Lipschitz-free space on MM is the closed convex hull of its strongly exposed points. Further, we prove that given a compact metric space MM which does not contain any isometric copy of [0,1][0,1] and a Banach space YY, if SNA(M,Y)(M,Y) is dense, then SNA(M,Y)(M,Y) 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. 1917–1951

DOI 10.4171/RMI/1253