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 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 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