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 vector space $(X,\|\cdot \|)$.

1. Let some ${\bar {x}}\in \operatorname {cl} (S)$ be a point in the closure of $S$. An element $h\in X$ is called a tangent (or tangent vector) 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 $\lambda _{n}>0$ such 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 ${\bar {x}}$.

An equivalent definition is given in terms of a distance function and the limit infimum. As before, let $(X,\|\cdot \|)$ be a normed vector space and take some nonempty set $S\subset X$. For each $x\in X$, let the distance function to $S$ be

$d_{S}(x):=\inf\{\|x-x'\|\mid x'\in S\}.$

Then, the contingent cone to $S\subset X$ at $x\in \operatorname {cl} (S)$ is defined by

$T_{S}(x):=\left\{v:\liminf _{h\to 0^{+}}{\frac {d_{S}(x+hv)}{h}}=0\right\}.$