Characterization of the Clarke regularity of subanalytic sets
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- by Abderrahim Jourani and Moustapha Séne PDF
- Proc. Amer. Math. Soc. 146 (2018), 1639-1649 Request permission
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 \[ \mathcal {L}(A, x_0):= \bigg \{\dot {c}_+(0) \in \mathbb {R}^n: \, c:[0,1]\longrightarrow A \;\; \text {is Lipschitz}, \; c(0)=x_0\bigg \}, \] 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.References
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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
- Email: abderrahim.jourani@u-bourgogne.fr
- Moustapha Séne
- Affiliation: Département de Mathématiques, Université Gaston Berger, Saint-Louis du Sénégal, Senegal
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 1639-1649
- MSC (2010): Primary 49J52, 46N10, 58C20; Secondary 34A60
- DOI: https://doi.org/10.1090/proc/13847
- MathSciNet review: 3754348