Paratingent cone

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

Definition

Let be a nonempty subset of a real normed vector space .

1. Let some be a point in the closure of . An element is called a tangent (or tangent vector) to at , if there is a sequence of elements and a sequence of positive real numbers such that and
2. The set of all tangents to at is called the contingent cone (or the Bouligand tangent cone) to at .

An equivalent definition is given in terms of a distance function and the limit infimum. As before, let be a normed vector space and take some nonempty set . For each , let the distance function to be

Then, the contingent cone to at is defined by

References