2019/07/17 by Rafael Chiclana, Chiclana, Rafael, Luis C. García‐Lirola +5 · 1 citation
Computer Science · Mathematics · #46B04 (Primary) #46B20 #46B22 #54E50 (Secondary) #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.1907.07698
openalex publication_date 2019/07/17 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
We study the density of the set \SNA(M,Y) of those Lipschitz\nmaps from a (complete pointed) metric space M to a Banach space Y which\nstrongly attain their norm (i.e. the supremum defining the Lipschitz norm is\nactually a maximum). We present new and somehow counterintuitive examples, and\nwe give some applications. First, we show that \SNA( mathbb\nT,Y) is not dense in \Lip0( mathbb T,Y) for any Banach space Y,\nwhere mathbb T denotes the unit circle in the Euclidean plane. This provides\nthe first example of a Gromov concave metric space (i.e. every molecule is a\nstrongly exposed point of the unit ball of the Lipschitz-free space) for which\nthe density does not hold. Next, we construct metric spaces M satisfying that\n\SNA(M,Y) is dense in \Lip0(M,Y) regardless Y\nbut which contains an isometric copy of [0,1] and so the Lipschitz-free space\n mathcal F(M) fails the Radon--Nikod 'ym property, answering in the\nnegative a posed question. Furthermore, an example M can be produced failing\nall the previously known sufficient conditions to get the density of strongly\nnorm attaining Lipschitz maps. Finally, among other applications, we prove that\ngiven a compact metric M which does not contains any isometric copy of\n[0,1] and a Banach space Y, if \SNA(M,Y) is dense, then\n\SNA(M,Y) actually contains an open dense subset and\nB mathcal F(M)=\\co(\str-exp(B mathcal\nF(M))). Further, we show that if M is a boundedly compact metric space for\nwhich \SNA(M, mathbb R) is dense in \Lip0(M, mathbb\nR), then the unit ball of the Lipschitz-free space on M is the closed convex\nhull of its strongly exposed points.\n