2024/04/11 by Geunsu Choi, Mingu Jung, Choi, Geunsu +5
Computer Science · Mathematics · #Optimization and Variational Analysis #Advanced Banach Space Theory #Advanced Topology and Set Theory
paper · pdf · doi:10.48550/arxiv.2404.07599
We solve two main questions on linear structures of (non-)norm-attaining Lipschitz functions. First, we show that for every infinite metric space M, the set consisting of Lipschitz functions on M which do not strongly attain their norm and the zero contains an isometric copy of ℓ_∞, and moreover, those functions can be chosen not to attain their norm as functionals on the Lipschitz-free space over M. Second, we prove that for every infinite metric space M, neither the set of strongly norm-attaining Lipschitz functions on M nor the union of its complement with zero is ever a linear space. Furthermore, we observe that the set consisting of Lipschitz functions which cannot be approximated by strongly norm-attaining ones and the zero element contains ℓ_∞ isometrically in all the known cases. Some natural observations and spaceability results are also investigated for Lipschitz functions that attain their norm in one way but do not in another, for several norm-attainment notions considered in the literature.