Algebraic interior explained

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.

Definition

Assume that

is a subset of a vector space

The algebraic interior (or radial kernel) of
A

with respect to
X

is the set of all points at which

is a radial set. A point

a₀\inA

is called an of

^[1] and

is said to be if for every

x\inX

there exists a real number

t_x>0

such that for every

t\in[0,t_x],

a₀+tx\inA.

This last condition can also be written as

a₀+[0,t_x]x\subseteqA

where the set

a_0 + [0, t_x] x ~:=~ \left\

is the line segment (or closed interval) starting at

a₀

and ending at

a₀+t_xx;

this line segment is a subset of

a₀+[0,infty)x,

which is the emanating from

a₀

in the direction of

(that is, parallel to/a translation of

[0,infty)x

). Thus geometrically, an interior point of a subset

is a point

a₀\inA

with the property that in every possible direction (vector)

x ≠ 0,

contains some (non-degenerate) line segment starting at

a₀

and heading in that direction (i.e. a subset of the ray

a₀+[0,infty)x

). The algebraic interior of

(with respect to

) is the set of all such points. That is to say, it is the subset of points contained in a given set with respect to which it is radial points of the set.^[2]

is a linear subspace of

and

A\subseteqX

then this definition can be generalized to the algebraic interior of
A

with respect to
M

is:

\operatorname_M A := \left\.

where

\operatorname{aint}_MA\subseteqA

always holds and if

\operatorname{aint}_MA ≠ \varnothing

then

M\subseteq\operatorname{aff}(A-A),

where

\operatorname{aff}(A-A)

is the affine hull of

A-A

(which is equal to

\operatorname{span}(A-A)

Algebraic closure

A point

x\inX

is said to be from a subset

A\subseteqX

if there exists some

a\inA

such that the line segment

[a,x):=a+[0,1)x

is contained in

The, denoted by

\operatorname{acl}_XA,

consists of

and all points in

that are linearly accessible from

Algebraic Interior (Core)

In the special case where

M:=X,

the set

\operatorname{aint}_XA

is called the or of
A

and it is denoted by

Aⁱ

\operatorname{core}A.

Formally, if

is a vector space then the algebraic interior of

A\subseteqX

is^[3]

\operatorname_X A := \operatorname(A) := \left\.

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):

$^ A := \begin^i A & \text \operatorname A \text \\\varnothing & \text\end$

$^ A := \begin^i A & \text \operatorname (A - a) \text X \text a \in A \text \\\varnothing & \text\end$

is a Fréchet space,

is convex, and

\operatorname{aff}A

is closed in

then

{}^icA={}^ibA

but in general it is possible to have

{}^icA=\varnothing

while

{}^ibA

is empty.

Examples

A=\{x\in\R^2:x₂\geq

	2
x
	1

orx₂\leq0\}\subseteq\R²

then

0\in\operatorname{core}(A),

but

0\not\in\operatorname{int}(A)

and

0\not\in\operatorname{core}(\operatorname{core}(A)).

Properties of core

Suppose

A,B\subseteqX.

In general,

\operatorname{core}A ≠ \operatorname{core}(\operatorname{core}A).

But if

is a convex set then:

\operatorname{core}A=\operatorname{core}(\operatorname{core}A),

and

- for all

x₀\in\operatorname{core}A,y\inA,0<λ\leq1

then

λx₀+(1-λ)y\in\operatorname{core}A.

is an absorbing subset of a real vector space if and only if

0\in\operatorname{core}(A).

A+\operatorname{core}B\subseteq\operatorname{core}(A+B)

A+\operatorname{core}B=\operatorname{core}(A+B)

B=\operatorname{core}B.

Both the core and the algebraic closure of a convex set are again convex. If

is convex,

c\in\operatorname{core}C,

and

b\in\operatorname{acl}_XC

then the line segment

[c,b):=c+[0,1)b

is contained in

\operatorname{core}C.

Relation to topological interior

Let

be a topological vector space,

\operatorname{int}

denote the interior operator, and

A\subseteqX

then:

\operatorname{int}A\subseteq\operatorname{core}A

is nonempty convex and

is finite-dimensional, then

\operatorname{int}A=\operatorname{core}A.

is convex with non-empty interior, then

\operatorname{int}A=\operatorname{core}A.

^[4]

is a closed convex set and

is a complete metric space, then

\operatorname{int}A=\operatorname{core}A.

^[5]

Relative algebraic interior

M=\operatorname{aff}(A-A)

then the set

\operatorname{aint}_MA

is denoted by

{}^iA:=\operatorname{aint}_{\operatorname{aff}(A-A)}A

and it is called the relative algebraic interior of
A.

This name stems from the fact that

a\inAⁱ

if and only if

\operatorname{aff}A=X

and

a\in{}^iA

(where

\operatorname{aff}A=X

if and only if

\operatorname{aff}(A-A)=X

Relative interior

is a subset of a topological vector space

then the relative interior of

is the set

\operatorname A := \operatorname_ A.

That is, it is the topological interior of A in

\operatorname{aff}A,

which is the smallest affine linear subspace of

containing

The following set is also useful:

\operatorname A := \begin\operatorname A & \text \operatorname A \text X \text \\\varnothing & \text\end

Quasi relative interior

is a subset of a topological vector space

then the quasi relative interior of

is the set

\operatorname A := \left\.

In a Hausdorff finite dimensional topological vector space,

\operatorname{qri}A={}ⁱA={}^icA={}^ibA.

References

Bibliography

Notes and References

Web site: Separation of Convex Sets in Linear Topological Spaces . November 14, 2012 . John Cook . May 21, 1988.
Web site: Coherent Risk Measures, Valuation Bounds, and (
\mu,\rho

)-Portfolio Optimization. Stefan. Jaschke. Uwe. Kuchler. 2000.
Book: Nikolaĭ Kapitonovich Nikolʹskiĭ. Functional analysis I: linear functional analysis. 1992. Springer. 978-3-540-50584-6.
Book: Kantorovitz, Shmuel. Introduction to Modern Analysis. Oxford University Press. 2003. 9780198526568. 134.
.