
TL;DR
This paper characterizes local Lipschitz one sets in finite-dimensional spaces, introduces a measure-theoretic quasi-density condition, and explores their properties and examples.
Contribution
It provides a measure-theoretic characterization of local Lipschitz one sets on the real line and analyzes their relation to regular closed sets in normed spaces.
Findings
Local Lipschitz one sets are quasi-dense but not all quasi-dense sets are local Lipschitz one.
Any regular closed subset of a normed space is a local Lipschitz one set.
Existence of local Lipschitz one sets that are not regular closed.
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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