Paratingent cone

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

Definition

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

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

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

[math]\displaystyle{ d_S(x) := \inf\{\|x - x'\| \mid x' \in S\}. }[/math]

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

[math]\displaystyle{ T_S(x) := \left\{v : \liminf_{h \to 0^+} \frac{d_S(x + hv)}{h} = 0 \right\}. }[/math]