Extreme non-differentiability of typical Lipschitz mappings
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917977705676800 |
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| author | Dymond, Michael Maleva, Olga |
| author_facet | Dymond, Michael Maleva, Olga |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04117 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extreme non-differentiability of typical Lipschitz mappings Dymond, Michael Maleva, Olga Functional Analysis 26B05, 46G05 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. |
| title | Extreme non-differentiability of typical Lipschitz mappings |
| topic | Functional Analysis 26B05, 46G05 |
| url | https://arxiv.org/abs/2504.04117 |