Generalization of topological interior
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] The elements of the algebraic interior are often referred to as internal points .[2] [3]
If
M
{\displaystyle M}
is a linear subspace of
X
{\displaystyle X}
and
A
⊆
X
{\displaystyle A\subseteq X}
then the algebraic interior of
A
{\displaystyle A}
with respect to
M
{\displaystyle M}
is:
aint
M
A
:=
{
a
∈
X
:
for all
m
∈
M
,
there exists some
t
m
>
0
such that
a
+
[
0
,
t
m
]
⋅
m
⊆
A
}
.
{\displaystyle \operatorname {aint} _{M}A:=\left\{a\in X:{\text{ for all }}m\in M,{\text{ there exists some }}t_{m}>0{\text{ such that }}a+\left[0,t_{m}\right]\cdot m\subseteq A\right\}.}
where it is clear that
aint
M
A
⊆
A
{\displaystyle \operatorname {aint} _{M}A\subseteq A}
and if
aint
M
A
≠
∅
{\displaystyle \operatorname {aint} _{M}A\neq \varnothing }
then
M
⊆
aff
(
A
−
A
)
{\displaystyle M\subseteq \operatorname {aff} (A-A)}
, where
aff
(
A
−
A
)
{\displaystyle \operatorname {aff} (A-A)}
is the
affine hull of
A
−
A
{\displaystyle A-A}
(which is equal to
span
(
A
−
A
)
{\displaystyle \operatorname {span} (A-A)}
).
Algebraic Interior (Core) [ ]
The set
aint
X
A
{\displaystyle \operatorname {aint} _{X}A}
is called the algebraic interior of
A
{\displaystyle A}
or the core of
A
{\displaystyle A}
and it is denoted by
A
i
{\displaystyle A^{i}}
or
core
A
{\displaystyle \operatorname {core} A}
.
Formally, if
X
{\displaystyle X}
is a vector space then the algebraic interior of
A
⊆
X
{\displaystyle A\subseteq X}
is
aint
X
A
:=
core
(
A
)
:=
{
a
∈
A
:
for all
x
∈
X
,
there exists some
t
x
>
0
,
such that for all
t
∈
[
0
,
t
x
]
,
a
+
t
x
∈
A
}
.
{\displaystyle \operatorname {aint} _{X}A:=\operatorname {core} (A):=\left\{a\in A:{\text{ for all }}x\in X,{\text{ there exists some }}t_{x}>0,{\text{ such that for all }}t\in \left[0,t_{x}\right],a+tx\in A\right\}.}
[5]
If
A
{\displaystyle A}
is non-empty, then these additional subsets are also useful for the statements of many theorems in convex functional analysis (such as the Ursescu theorem ):
i
c
A
:=
{
i
A
if
aff
A
is a closed set,
∅
otherwise
{\displaystyle {}^{ic}A:={\begin{cases}{}^{i}A&{\text{ if }}\operatorname {aff} A{\text{ is a closed set,}}\\\varnothing &{\text{ otherwise}}\end{cases}}}
i
b
A
:=
{
i
A
if
span
(
A
−
a
)
is a barrelled linear subspace of
X
for any/all
a
∈
A
,
∅
otherwise
{\displaystyle {}^{ib}A:={\begin{cases}{}^{i}A&{\text{ if }}\operatorname {span} (A-a){\text{ is a barrelled linear subspace of }}X{\text{ for any/all }}a\in A{\text{,}}\\\varnothing &{\text{ otherwise}}\end{cases}}}
If
X
{\displaystyle X}
is a Fréchet space ,
A
{\displaystyle A}
is convex, and
aff
A
{\displaystyle \operatorname {aff} A}
is closed in
X
{\displaystyle X}
then
i
c
A
=
i
b
A
{\displaystyle {}^{ic}A={}^{ib}A}
but in general it's possible to have
i
c
A
=
∅
{\displaystyle {}^{ic}A=\varnothing }
while
i
b
A
{\displaystyle {}^{ib}A}
is not empty.
Example [ ]
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}}
then
0
∈
core
(
A
)
{\displaystyle 0\in \operatorname {core} (A)}
, but
0
∉
int
(
A
)
{\displaystyle 0\not \in \operatorname {int} (A)}
and
0
∉
core
(
core
(
A
)
)
.
{\displaystyle 0\not \in \operatorname {core} (\operatorname {core} (A)).}
Properties of core [ ]
If
A
,
B
⊆
X
{\displaystyle A,B\subseteq X}
then:
In general,
core
(
A
)
≠
core
(
core
(
A
)
)
.
{\displaystyle \operatorname {core} (A)\neq \operatorname {core} (\operatorname {core} (A)).}
If
A
{\displaystyle A}
is a convex set then:
core
(
A
)
=
core
(
core
(
A
)
)
,
{\displaystyle \operatorname {core} (A)=\operatorname {core} (\operatorname {core} (A)),}
and
for all
x
0
∈
core
A
,
y
∈
A
,
0
<
λ
≤
1
{\displaystyle x_{0}\in \operatorname {core} A,y\in A,0<\lambda \leq 1}
then
λ
x
0
+
(
1
−
λ
)
y
∈
core
A
{\displaystyle \lambda x_{0}+(1-\lambda )y\in \operatorname {core} A}
A
{\displaystyle A}
is 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\subseteq \operatorname {core} (A+B)}
[6]
A
+
core
B
=
core
(
A
+
B
)
{\displaystyle A+\operatorname {core} B=\operatorname {core} (A+B)}
if
B
=
core
B
.
{\displaystyle B=\operatorname {core} B.}
[6]
Relation to interior [ ]
Let
X
{\displaystyle X}
be a topological vector space ,
int
{\displaystyle \operatorname {int} }
denote the interior operator, and
A
⊆
X
{\displaystyle A\subseteq X}
then:
int
A
⊆
core
A
{\displaystyle \operatorname {int} A\subseteq \operatorname {core} A}
If
A
{\displaystyle A}
is nonempty convex and
X
{\displaystyle X}
is finite-dimensional, then
int
A
=
core
A
.
{\displaystyle \operatorname {int} A=\operatorname {core} A.}
[2]
If
A
{\displaystyle A}
is convex with non-empty interior, then
int
A
=
core
A
.
{\displaystyle \operatorname {int} A=\operatorname {core} A.}
[7]
If
A
{\displaystyle A}
is a closed convex set and
X
{\displaystyle X}
is a complete metric space , then
int
A
=
core
A
.
{\displaystyle \operatorname {int} A=\operatorname {core} A.}
[8]
Relative algebraic interior [ ]
If
M
=
aff
(
A
−
A
)
{\displaystyle M=\operatorname {aff} (A-A)}
then the set
aint
M
A
{\displaystyle \operatorname {aint} _{M}A}
is denoted by
i
A
:=
aint
aff
(
A
−
A
)
A
{\displaystyle {}^{i}A:=\operatorname {aint} _{\operatorname {aff} (A-A)}A}
and it is called the relative algebraic interior of
A
.
{\displaystyle A.}
[6] This name stems from the fact that
a
∈
A
i
{\displaystyle a\in A^{i}}
if and only if
aff
A
=
X
{\displaystyle \operatorname {aff} A=X}
and
a
∈
i
A
{\displaystyle a\in {}^{i}A}
(where
aff
A
=
X
{\displaystyle \operatorname {aff} A=X}
if and only if
aff
(
A
−
A
)
=
X
{\displaystyle \operatorname {aff} (A-A)=X}
).
Relative interior [ ]
If
A
{\displaystyle A}
is a subset of a topological vector space
X
{\displaystyle X}
then the relative interior of
A
{\displaystyle A}
is the set
rint
A
:=
int
aff
A
A
.
{\displaystyle \operatorname {rint} A:=\operatorname {int} _{\operatorname {aff} A}A.}
That is, it is the topological interior of A in
aff
A
{\displaystyle \operatorname {aff} A}
, which is the smallest affine linear subspace of
X
{\displaystyle X}
containing
A
.
{\displaystyle A.}
The following set is also useful:
ri
A
:=
{
rint
A
if
aff
A
is a closed subspace of
X
,
∅
otherwise
{\displaystyle \operatorname {ri} A:={\begin{cases}\operatorname {rint} A&{\text{ if }}\operatorname {aff} A{\text{ is a closed subspace of }}X{\text{,}}\\\varnothing &{\text{ otherwise}}\end{cases}}}
Quasi relative interior [ ]
If
A
{\displaystyle A}
is a subset of a topological vector space
X
{\displaystyle X}
then the quasi relative interior of
A
{\displaystyle A}
is the set
qri
A
:=
{
a
∈
A
:
cone
¯
(
A
−
a
)
is a linear subspace of
X
}
.
{\displaystyle \operatorname {qri} A:=\left\{a\in A:{\overline {\operatorname {cone} }}(A-a){\text{ is a linear subspace of }}X\right\}.}
In a Hausdorff finite dimensional topological vector space,
qri
A
=
i
A
=
i
c
A
=
i
b
A
.
{\displaystyle \operatorname {qri} A={}^{i}A={}^{ic}A={}^{ib}A.}
See also [ ]
Bounding point – Mathematical concept related to subsets of vector spaces
Interior (topology) – Largest open subset of some given set
Quasi-relative interior – Generalization of algebraic interior
Relative interior – Generalization of topological interior
Order unit – Element of an ordered vector space
Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem
References [ ]
^ a b Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and (
μ
,
ρ
{\displaystyle \mu ,\rho }
)-Portfolio Optimization".
^ 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 c 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 , Springer series in operations research, Springer, Remark 2.73, p. 56, ISBN 9780387987057 .
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