Characterization of the Clarke regularity of subanalytic sets

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's tangent cone to $A$ at $x_0$ with the set \\ $$\mathcal{L}(A, x_0):= \bigg\{\dot{c}_+(0) \in \mathbb{R}^n: \, c:[0,1]\longrightarrow A\;\;\mbox{\it is Lipschitz}, \, c(0)=x_0\bigg\}.$$ 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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https://hal.archives-ouvertes.fr/hal-01666603
Contributor : Abderrahim Jourani <>
Submitted on : Monday, December 18, 2017 - 3:35:58 PM
Last modification on : Friday, October 19, 2018 - 11:32:01 AM

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Abderrahim Jourani, Moustapha Séne. Characterization of the Clarke regularity of subanalytic sets. Proceedings of the American Mathematical Society, American Mathematical Society, 2018, 146 (4), pp.1639-1649. ⟨10.1090/proc/13847⟩. ⟨hal-01666603⟩

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