Paratingent cone

In mathematics, the paratingent cone and contingent cone were introduced by, 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
 * $$ T_S(x) := \left\{v : \liminf_{h \to 0^+} \frac{d_S(x + hv)}{h} = 0 \right\}. $$