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Abstract

We study two types of approximations of Lipschitz maps with derivatives of maximal slopes on Banach spaces. First, we characterize the Radon–Nikodým property in terms of strongly norm attaining Lipschitz maps and maximal derivative attaining Lipschitz maps, which complements the characterization presented by Choi et al. (2020). It is shown in particular that if every Lipschitz map can be approximated by those that either strongly attain their norm or attain their maximal derivative for every renorming of the range space, then the range space must have the Radon–Nikodým property. Next, we prove that every Lipschitz functional defined on the real line can be locally approximated by maximal affine functions, while uniform approximation cannot be guaranteed. This extends the previous work of Bates et al. (1999) from the perspective of maximal affine functions.