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Clarke's tangent cones and the boundaries of closed sets in Rn

1979; Elsevier BV; Volume: 3; Issue: 1 Linguagem: Inglês

10.1016/0362-546x(79)90044-0

1873-5215

R. T. Rockafellar,

Point processes and geometric inequalities

Let $C$ be a nonempty closed subset of $\mathbb{R}^n$. For each $x \in C$, the tangent cone $T_C(x)$ in the sense of Clarke consists of all $y \in \mathbb{R}^n$ such that, whenever one has sequences $t_k\downarrow 0$ and $x_k \rightarrow x$ with $x_k \in C$, there exist $y_k \rightarrow y$ with $x_k + t_ky_k \in C$ for all $k$. This is not Clarke’s original definition but it is equivalent to it.