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(T, Y ) is not dense in Lip 0 (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 Lip 0 (M, Y ) regardless Y but which contains an isometric copy of [0, 1] and so the Lipschitz-free space 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 SNA(M, Y ) is dense, then SNA(M, Y ) actually contains an open dense subset and B F (M ) = co(str-exp B F (M ) ). Further, we show that if M is a boundedly compact metric space for which SNA(M, R) is dense in Lip 0 (M, R), then the unit ball of the Lipschitz-free space on M is the closed convex hull of its strongly exposed points.