algebraic 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.

Definition

Assume that $A$ is a subset of a vector space $X.$

The algebraic interior (or radial kernel) of $A$ with respect to $X$ is the set of all points at which $A$ is a radial set.

A point $a_0 \in A$ is called an {{em|internal point}} of $A$ {{sfn|Aliprantis|Border|2006|pp=199–200}}{{cite web|url=http://www.johndcook.com/SeparationOfConvexSets.pdf | access-date=November 14, 2012 |title=Separation of Convex Sets in Linear Topological Spaces |author=John Cook |date=May 21, 1988}} and $A$ is said to be {{em|radial at $a_0$ }} if for every $x \in X$ there exists a real number $t_x > 0$ such that for every $t \in [0, t_x],$ $a_0 + t x \in A.$

This last condition can also be written as $a_0 + [0, t_x] x \subseteq A$ where the set

$a_0 + [0, t_x] x ~:=~ \left\{a_0 + t x : t \in [0, t_x]\right\}$

is the line segment (or closed interval) starting at $a_0$ and ending at $a_0 + t_x x;$

this line segment is a subset of $a_0 + [0, \infty) x,$ which is the {{em|ray}} emanating from $a_0$ in the direction of $x$ (that is, parallel to/a translation of $[0, \infty) x$ ).

Thus geometrically, an interior point of a subset $A$ is a point $a_0 \in A$ with the property that in every possible direction (vector) $x \neq 0,$ $A$ contains some (non-degenerate) line segment starting at $a_0$ and heading in that direction (i.e. a subset of the ray $a_0 + [0, \infty) x$ ).

The algebraic interior of $A$ (with respect to $X$ ) 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.{{cite web|title=Coherent Risk Measures, Valuation Bounds, and ( $\mu,\rho$ )-Portfolio Optimization|first1=Stefan|last1=Jaschke|first2=Uwe|last2=Kuchler|date=2000|url=https://edoc.hu-berlin.de/bitstream/handle/18452/4328/64.pdf?sequence=1}}

If $M$ is a linear subspace of $X$ and $A \subseteq X$ then this definition can be generalized to the algebraic interior of $A$ with respect to $M$ is:{{sfn|Zălinescu|2002|p=2}}

$\operatorname{aint}_M A := \left\{ a \in X : \text{ for all } m \in M, \text{ there exists some } t_m > 0 \text{ such that } a + \left[0, t_m\right] \cdot m \subseteq A \right\}.$

where $\operatorname{aint}_M A \subseteq A$ always holds and if $\operatorname{aint}_M A \neq \varnothing$ 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 $x \in X$ is said to be {{em|{{visible anchor|linearly accessible}}}} from a subset $A \subseteq X$ if there exists some $a \in A$ such that the line segment $[a, x) := a + [0, 1) (x-a)$ is contained in $A.$ {{sfn|Narici|Beckenstein|2011|p=109}}

The algebraic closure of $A$ with respect to $X$ , denoted by $\operatorname{acl}_X A,$ consists of ( $A$ and) all points in $X$ that are linearly accessible from $A.$ {{sfn|Narici|Beckenstein|2011|p=109}}

Algebraic Interior (Core)

In the special case where $M := X,$ the set $\operatorname{aint}_X A$ is called the {{visible anchor|algebraic interior}} or {{visible anchor|core}} of $A$ and it is denoted by $A^i$ or $\operatorname{core} A.$

Formally, if $X$ is a vector space then the algebraic interior of $A \subseteq X$ is{{cite book|author=Nikolaĭ Kapitonovich Nikolʹskiĭ|title=Functional analysis I: linear functional analysis|date=1992|publisher=Springer|isbn=978-3-540-50584-6}}

$\operatorname{aint}_X A := \operatorname{core}(A) := \left\{ a \in A : \text{ for all } x \in X, \text{ there exists some } t_x > 0, \text{ such that for all } t \in \left[0, t_x\right], a + tx \in A \right\}.$

We call A algebraically open in X if $A = \operatorname{aint}_X A$

If $A$ 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):

${}^{ic} A :=
\begin{cases}
{}^i A & \text{ if } \operatorname{aff} A \text{ is a closed set,} \\
\varnothing & \text{ otherwise}
\end{cases}$

${}^{ib} A :=
\begin{cases}
{}^i A & \text{ if } \operatorname{span} (A - a) \text{ is a barrelled linear subspace of } X \text{ for any/all } a \in A \text{,} \\
\varnothing & \text{ otherwise}
\end{cases}$

If $X$ is a Fréchet space, $A$ is convex, and $\operatorname{aff} A$ is closed in $X$ then ${}^{ic} A = {}^{ib} A$ but in general it is possible to have ${}^{ic} A = \varnothing$ while ${}^{ib} A$ is {{em|not}} empty.

=Examples=

If $A = \{x \in \R^2: x_2 \geq x_1^2 \text{ or } x_2 \leq 0\} \subseteq \R^2$ 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 $A, B \subseteq X.$

In general, $\operatorname{core} A \neq \operatorname{core}(\operatorname{core} A).$ But if $A$ is a convex set then:
$\operatorname{core} A = \operatorname{core}(\operatorname{core} A),$ and
for all $x_0 \in \operatorname{core} A, y \in A, 0 < \lambda \leq 1$ then $\lambda x_0 + (1 - \lambda)y \in \operatorname{core} A.$
$A$ is an absorbing subset of a real vector space if and only if $0 \in \operatorname{core}(A).$
$A + \operatorname{core} B \subseteq \operatorname{core}(A + B)$ {{sfn|Zălinescu|2002|pp=2–3}}
$A + \operatorname{core} B = \operatorname{core}(A + B)$ if $B = \operatorname{core}B.$ {{sfn|Zălinescu|2002|pp=2–3}}

Both the core and the algebraic closure of a convex set are again convex.{{sfn|Narici|Beckenstein|2011|p=109}}

If $C$ is convex, $c \in \operatorname{core} C,$ and $b \in \operatorname{acl}_X C$ then the line segment $[c, b) := c + [0, 1) b$ is contained in $\operatorname{core} C.$ {{sfn|Narici|Beckenstein|2011|p=109}}

=Relation to topological interior=

Let $X$ be a topological vector space, $\operatorname{int}$ denote the interior operator, and $A \subseteq X$ then:

$\operatorname{int}A \subseteq \operatorname{core}A$
If $A$ is nonempty convex and $X$ is finite-dimensional, then $\operatorname{int} A = \operatorname{core} A.$ {{sfn|Aliprantis|Border|2006|pp=199–200}}
If $A$ is convex with non-empty interior, then $\operatorname{int}A = \operatorname{core} A.$ {{cite book|last=Kantorovitz|first=Shmuel|title=Introduction to Modern Analysis|publisher=Oxford University Press|date=2003|isbn=9780198526568|page=134}}
If $A$ is a closed convex set and $X$ is a complete metric space, then $\operatorname{int} A = \operatorname{core} A.$ {{citation|last1=Bonnans|first1=J. Frederic|last2=Shapiro|first2=Alexander|title=Perturbation Analysis of Optimization Problems|series=Springer series in operations research|publisher=Springer|date=2000|isbn=9780387987057|at=Remark 2.73, p. 56|url=https://books.google.com/books?id=ET70F9HgIpIC&pg=PA56}}.

Relative algebraic interior

If $M = \operatorname{aff} (A - A)$ then the set $\operatorname{aint}_M A$ is denoted by ${}^iA := \operatorname{aint}_{\operatorname{aff} (A - A)} A$ and it is called the relative algebraic interior of $A.$ {{sfn|Zălinescu|2002|pp=2–3}} This name stems from the fact that $a \in A^i$ 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

If $A$ is a subset of a topological vector space $X$ then the relative interior of $A$ is the set

$\operatorname{rint} A := \operatorname{int}_{\operatorname{aff} A} A.$

That is, it is the topological interior of A in $\operatorname{aff} A,$ which is the smallest affine linear subspace of $X$ containing $A.$ The following set is also useful:

$\operatorname{ri} A :=
\begin{cases}
\operatorname{rint} A & \text{ if } \operatorname{aff} A \text{ is a closed subspace of } X \text{,} \\
\varnothing & \text{ otherwise}
\end{cases}$

Quasi relative interior

If $A$ is a subset of a topological vector space $X$ then the quasi relative interior of $A$ is the set

$\operatorname{qri} A := \left\{ a \in A : \overline{\operatorname{cone}} (A - a) \text{ is a linear subspace of } X \right\}.$

In a Hausdorff finite dimensional topological vector space, $\operatorname{qri} A = {}^i A = {}^{ic} A = {}^{ib} A.$

References

=Bibliography=

{{Aliprantis Border Infinite Dimensional Analysis A Hitchhiker's Guide Third Edition}}
{{Narici Beckenstein Topological Vector Spaces|edition=2}}
{{Schaefer Wolff Topological Vector Spaces|edition=2}}
{{Schechter Handbook of Analysis and Its Foundations}}
{{Zălinescu Convex Analysis in General Vector Spaces 2002}}

Category:Convex analysis

Category:Functional analysis

Category:Mathematical analysis

Category:Topology