Paratingent cone

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

Definition

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

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

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

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

Then, the contingent cone to ${\displaystyle S\subset X}$ at ${\displaystyle x\in \operatorname {cl} (S)}$ is defined by^[2]

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