On local Lipschitz one sets
Functional Analysis
2026-04-22 v1
Abstract
We study the local Lipschitz one subsets of a finite dimensional space, that is, sets for which there exists a continuous function whose local Lipschitz derivative is the characteristic function of said set. We give a characterization of a local Lipschitz one set on the real line in terms of a certain measure-theoretic density condition, which we call quasi-density. We show that any local Lipschitz one set needs to be quasi-dense, but the converse does not hold. Finally, we show that any regular closed subset of a normed space is a local Lipschitz one set, but there exist local Lipschitz one sets that are not regular closed.
Cite
@article{arxiv.2604.19704,
title = {On local Lipschitz one sets},
author = {Ziemowit M. Wójcicki},
journal= {arXiv preprint arXiv:2604.19704},
year = {2026}
}