Extreme non-differentiability of typical Lipschitz mappings
Functional Analysis
2025-04-08 v1
Abstract
We show that no matter what subset of a normed space is given, a typical 1-Lipschitz mapping into a Banach space is non-differentiable at a typical point of the set in a very strong sense: the derivative ratio approximates, on arbitrary small scales, every linear operator of norm at most 1. For subsets of finite-dimensional normed spaces which can be covered by a countable union of closed purely unrectifiable sets this extreme non-differentiability holds for a typical Lipschitz mapping at every point. Both results are new even for Lipschitz mappings with a finite-dimensional co-domain.
Cite
@article{arxiv.2504.04117,
title = {Extreme non-differentiability of typical Lipschitz mappings},
author = {Michael Dymond and Olga Maleva},
journal= {arXiv preprint arXiv:2504.04117},
year = {2025}
}
Comments
This paper supersedes our preprint arXiv:2111.09644 [math.FA]. Results from arXiv:2111.09644 are incorporated in Sections 2 and 3 of the present paper