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In mathematics, the paratingent cone and contingent cone were introduced by
Bouligand (
1932 ), and are closely related to
tangent cones .
Definition
Let
S
{\displaystyle S}
be a nonempty subset of a
real
normed vector space
(
X
,
‖
⋅
‖
)
{\displaystyle (X,\|\cdot \|)}
.
Let some
x
¯
∈
cl
(
S
)
{\displaystyle {\bar {x}}\in \operatorname {cl} (S)}
be a point in the
closure of
S
{\displaystyle S}
. An element
h
∈
X
{\displaystyle h\in X}
is called a tangent (or tangent vector ) to
S
{\displaystyle S}
at
x
¯
{\displaystyle {\bar {x}}}
, if there is a sequence
(
x
n
)
n
∈
N
{\displaystyle (x_{n})_{n\in \mathbb {N} }}
of elements
x
n
∈
S
{\displaystyle x_{n}\in S}
and a sequence
(
λ
n
)
n
∈
N
{\displaystyle (\lambda _{n})_{n\in \mathbb {N} }}
of positive real numbers
λ
n
>
0
{\displaystyle \lambda _{n}>0}
such that
x
¯
=
lim
n
→
∞
x
n
{\displaystyle {\bar {x}}=\lim _{n\to \infty }x_{n}}
and
h
=
lim
n
→
∞
λ
n
(
x
n
−
x
¯
)
.
{\displaystyle h=\lim _{n\to \infty }\lambda _{n}(x_{n}-{\bar {x}}).}
The set
T
(
S
,
x
¯
)
{\displaystyle T(S,{\bar {x}})}
of all tangents to
S
{\displaystyle S}
at
x
¯
{\displaystyle {\bar {x}}}
is called the contingent cone (or the Bouligand tangent cone ) to
S
{\displaystyle S}
at
x
¯
{\displaystyle {\bar {x}}}
.
[1]
An equivalent definition is given in terms of a distance function and the limit infimum.
As before, let
(
X
,
‖
⋅
‖
)
{\displaystyle (X,\|\cdot \|)}
be a normed vector space and take some nonempty set
S
⊂
X
{\displaystyle S\subset X}
. For each
x
∈
X
{\displaystyle x\in X}
, let the distance function to
S
{\displaystyle S}
be
d
S
(
x
)
:=
inf
{
‖
x
−
x
′
‖
∣
x
′
∈
S
}
.
{\displaystyle d_{S}(x):=\inf\{\|x-x'\|\mid x'\in S\}.}
Then, the contingent cone to
S
⊂
X
{\displaystyle S\subset X}
at
x
∈
cl
(
S
)
{\displaystyle x\in \operatorname {cl} (S)}
is defined by
[2]
T
S
(
x
)
:=
{
v
:
lim inf
h
→
0
+
d
S
(
x
+
h
v
)
h
=
0
}
.
{\displaystyle T_{S}(x):=\left\{v:\liminf _{h\to 0^{+}}{\frac {d_{S}(x+hv)}{h}}=0\right\}.}
References