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Characterization of the Clarke regularity of subanalytic sets

Authors: Abderrahim Jourani and Moustapha Séne
Journal: Proc. Amer. Math. Soc. 146 (2018), 1639-1649
MSC (2010): Primary 49J52, 46N10, 58C20; Secondary 34A60
Published electronically: November 7, 2017
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Abstract: In this note, we will show that for a closed subanalytic subset $ A \subset \mathbb{R}^n$, the Clarke tangential regularity of $ A$ at $ x_0 \in A$ is equivalent to the coincidence of the Clarke tangent cone to $ A$ at $ x_0$ with the set

$\displaystyle \mathcal {L}(A, x_0):= \bigg \{\dot {c}_+(0) \in \mathbb{R}^n: \, c:[0,1]\longrightarrow A\;\;$$\displaystyle \mbox {\it is Lipschitz}, \, c(0)=x_0\bigg \},$

where $ \dot {c}_+(0)$ denotes the right-strict derivative of $ c$ at 0. The results obtained are used to show that the Clarke regularity of the epigraph of a function may be characterized by a new formula of the Clarke subdifferential of that function.

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Additional Information

Abderrahim Jourani
Affiliation: Université de Bourgogne Franche-Comté, Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, 21078 Dijon CEDEX, France

Moustapha Séne
Affiliation: Département de Mathématiques, Université Gaston Berger, Saint-Louis du Sénégal, Senegal

Keywords: Tangent cone, Clarke regularity, subanalytic set.
Received by editor(s): December 25, 2016
Received by editor(s) in revised form: May 31, 2017
Published electronically: November 7, 2017
Communicated by: Mourad Ismail
Article copyright: © Copyright 2017 American Mathematical Society

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