In functional analysis , a branch of mathematics, the algebraic interior or radial kernel of a subset of a vector space is a refinement of the concept of the interior . It is the subset of points contained in a given set with respect to which it is absorbing , i.e. the radial points of the set.[1] internal points .[2] [3]
Formally, if
X
{\displaystyle X}
linear space then the algebraic interior of
A
⊆
X
{\displaystyle A\subseteq X}
core
(
A
)
:=
{
x
0
∈
A
:
∀
x
∈
X
,
∃
t
x
>
0
,
∀
t
∈
[
0
,
t
x
]
,
x
0
+
t
x
∈
A
}
.
{\displaystyle \operatorname {core} (A):=\left\{x_{0}\in A:\forall x\in X,\exists t_{x}>0,\forall t\in [0,t_{x}],x_{0}+tx\in A\right\}.}
[4] Note that in general
core
(
A
)
≠
core
(
core
(
A
)
)
{\displaystyle \operatorname {core} (A)\neq \operatorname {core} (\operatorname {core} (A))}
A
{\displaystyle A}
convex set then
core
(
A
)
=
core
(
core
(
A
)
)
{\displaystyle \operatorname {core} (A)=\operatorname {core} (\operatorname {core} (A))}
A
{\displaystyle A}
x
0
∈
core
(
A
)
,
y
∈
A
,
0
<
λ
≤
1
{\displaystyle x_{0}\in \operatorname {core} (A),y\in A,0<\lambda \leq 1}
λ
x
0
+
(
1
−
λ
)
y
∈
core
(
A
)
{\displaystyle \lambda x_{0}+(1-\lambda )y\in \operatorname {core} (A)}
Example [ edit ] If
A
=
{
x
∈
R
2
:
x
2
≥
x
1
2
or
x
2
≤
0
}
⊆
R
2
{\displaystyle A=\{x\in \mathbb {R} ^{2}:x_{2}\geq x_{1}^{2}{\text{ or }}x_{2}\leq 0\}\subseteq \mathbb {R} ^{2}}
0
∈
core
(
A
)
{\displaystyle 0\in \operatorname {core} (A)}
0
∉
int
(
A
)
{\displaystyle 0\not \in \operatorname {int} (A)}
0
∉
core
(
core
(
A
)
)
{\displaystyle 0\not \in \operatorname {core} (\operatorname {core} (A))}
Properties [ edit ] Let
A
,
B
⊂
X
{\displaystyle A,B\subset X}
A
{\displaystyle A}
absorbing if and only if
0
∈
core
(
A
)
{\displaystyle 0\in \operatorname {core} (A)}
[1]
A
+
core
B
⊂
core
(
A
+
B
)
{\displaystyle A+\operatorname {core} B\subset \operatorname {core} (A+B)}
[5]
A
+
core
B
=
core
(
A
+
B
)
{\displaystyle A+\operatorname {core} B=\operatorname {core} (A+B)}
B
=
core
B
{\displaystyle B=\operatorname {core} B}
[5] Relation to interior [ edit ] Let
X
{\displaystyle X}
topological vector space ,
int
{\displaystyle \operatorname {int} }
A
⊂
X
{\displaystyle A\subset X}
int
A
⊆
core
A
{\displaystyle \operatorname {int} A\subseteq \operatorname {core} A}
If
A
{\displaystyle A}
X
{\displaystyle X}
int
A
=
core
A
{\displaystyle \operatorname {int} A=\operatorname {core} A}
[2]
If
A
{\displaystyle A}
int
A
=
core
A
{\displaystyle \operatorname {int} A=\operatorname {core} A}
[6]
If
A
{\displaystyle A}
X
{\displaystyle X}
complete metric space , then
int
A
=
core
A
{\displaystyle \operatorname {int} A=\operatorname {core} A}
[7] See also [ edit ] References [ edit ]
^ a b Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and (
μ
,
ρ
{\displaystyle \mu ,\rho }
^ a b Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3rd ed.). Springer. pp. 199–200. doi :10.1007/3-540-29587-9 . ISBN 978-3-540-32696-0 . ^ John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (pdf) . Retrieved November 14, 2012 . ^ Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis . Springer. ISBN 978-3-540-50584-6 . ^ a b Zălinescu, C. (2002). Convex analysis in general vector spaces . River Edge, NJ: World Scientific Publishing Co., Inc. pp. 2–3. ISBN 981-238-067-1 . MR 1921556 . ^ Shmuel Kantorovitz (2003). Introduction to Modern Analysis . Oxford University Press . p. 134. ISBN 9780198526568 . ^ Bonnans, J. Frederic; Shapiro, Alexander (2000), Perturbation Analysis of Optimization Problems ISBN 9780387987057
![\operatorname {core}(A):=\left\{x_{0}\in A:\forall x\in X,\exists t_{x}>0,\forall t\in [0,t_{x}],x_{0}+tx\in A\right\}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ad783b2516aa4afe53659ee2105cc18ac096401)
