Any local Lipschitz one set on the real line must be quasi-dense but not conversely, and every regular closed subset of a normed space is a local Lipschitz one set though the converse fails.
An Approach Based on Upper Gra- dients.Cambridge University Press
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On local Lipschitz one sets
Any local Lipschitz one set on the real line must be quasi-dense but not conversely, and every regular closed subset of a normed space is a local Lipschitz one set though the converse fails.