Paratingent cone explained

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, \| ⋅ \|) .

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} of elements x _{n} \in S and a sequence ( λ _{n)} _{n\inN} of positive real numbers λ _{n} > 0 such that \bar{x} = \lim _{n} x _{n} and h = \lim _{n} λ _{n} (x _{n} - \bar{x}).

1. 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} .^[1]

An equivalent definition is given in terms of a distance function and the limit infimum.As before, let

(X, \| ⋅ \|) 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^[2] T _{S(x)} : = \left\{v : \liminf _{h \to 0 ^{+}} (d _{S(x} + hv))/(h) = 0 \right\}.

Notes and References

1. Book: Johannes . Jahn . 2011 . Vector Optimization . Springer Berlin Heidelberg . 90–91 . 10.1007/978-3-642-17005-8 . 978-3-642-17005-8.
2. Book: Aubin . Jean-Pierre . Frankowska . Hèléne . 2009 . Chapter 4: Tangent Cones . Set-Valued Analysis . Modern Birkhäuser Classics . Boston . Birkhäuser . 2009 . 121 . 10.1007/978-0-8176-4848-0_4 . 978-0-8176-4848-0.