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arXiv · 2604.19704

On local Lipschitz one sets

Ziemowit M. Wójcicki
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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.

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Ziemowit M. Wójcicki. 2026-04-21. On local Lipschitz one sets. https://arxiv.org/abs/2604.19704

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