Norm attaining Lipschitz maps toward vectors

Choi, Geunsu

doi:10.1090/proc/16284

Norm attaining Lipschitz maps toward vectors
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by Geunsu Choi

Proc. Amer. Math. Soc. 151 (2023), 1729-1741

DOI: https://doi.org/10.1090/proc/16284

Published electronically: January 26, 2023

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Abstract:

We characterize the norm attainment toward vectors for vector-valued Lipschitz maps defined on a general metric space. The main theorem of the present paper states that on a large class of metric spaces including infinite subsets of finite-dimensional spaces, every Lipschitz map attains its norm toward a vector if and only if the range space is finite-dimensional. Furthermore, motivated by the first negative example given by G. Godefroy, some denseness results for norm attaining Lipschitz maps toward vectors are also presented.

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