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 $(X,\Vert \cdot \Vert )$.

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

An equivalent definition is given in terms of a distance function and the limit infimum. As before, let $(X,\Vert \cdot \Vert )$ 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

$d_S(x):=\inf \{\Vert x-x^{\prime }\Vert \mid x^{\prime }\in S\}.$

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

$T_S(x):=\{v:\underset{h\to 0^{+}}{\mathrm{lim\; inf}}\frac{d_S(x+hv)}{h}=0\}.$