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

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Definition

Let

S

be a nonempty subset of a

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(X, \, \cdot\, )

. # Let some

\bar \in \operatorname(S)

be a point in the closure of

S

. An element

h \in X

is called a ''tangent'' (or ''tangent vector'') to

S

\bar

, if there is a sequence

(x_n)_

of elements

x_n \in S

and a sequence

(\lambda_n)_

of positive real numbers

\lambda_n > 0

such that

\bar =  \lim_ x_n

and

h = \lim_ \lambda_n (x_n - \bar).

# The set

T(S,\bar)

of all tangents to

S

\bar

is called the ''contingent cone'' (or the ''Bouligand tangent cone'') to

S

\bar

. 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\.

Then, the ''contingent cone'' to

S \subset X

x \in \operatorname(S)

is defined by :

T_S(x) := \left\.

References

Mathematical analysis {{Mathanalysis-stub