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 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.