On local Lipschitz one sets

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.