Paratingent cone

In mathematics, the paratingent cone and contingent cone were introduced by Bouligand (1932), and are closely related to tangent cones.

Definition

Let $S$ be a nonempty subset of a real normed space $(X,\|\cdot \|)$.

1. Let some ${\bar {x}}\in \operatorname {cl} (S)$ be given. An element $h\in X$ is called a tangent to $S$ at ${\bar {x}}$, if there is a sequence $(x_{n})_{n\in \mathbb {N} }$ of elements $x_{n}\in S$ and a sequence $(\lambda _{n})_{n\in \mathbb {N} }$ of positive real numbers $\lambda _{n}>0$ so that ${\bar {x}}=\lim _{n\to \infty }x_{n}$ and $h=\lim _{n\to \infty }\lambda _{n}(x_{n}-{\bar {x}}).$
2. The set $T(S,{\bar {x}})$ of all tangents to $S$ at ${\bar {x}}$ is called the contingent cone (or the Bouligand tangent cone) to $S$ at ${\bar {x}}$.