View source for Algebraic interior

{{Short description|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 (topology)|interior]].

==Definition==

Assume that <math>A</math> is a subset of a vector space <math>X.</math> 
The ''algebraic interior'' (or ''radial kernel'') ''of <math>A</math> with respect to <math>X</math>'' is the set of all points at which <math>A</math> is a [[radial set]]. 
A point <math>a_0 \in A</math> is called an {{em|internal point}} of <math>A</math>{{sfn|Aliprantis|Border|2006|pp=199–200}}<ref name="cook">{{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}}</ref> and <math>A</math> is said to be {{em|[[Radial set|radial]] at <math>a_0</math>}} if for every <math>x \in X</math> there exists a real number <math>t_x > 0</math> such that for every <math>t \in [0, t_x],</math> <math>a_0 + t x \in A.</math> 
This last condition can also be written as <math>a_0 + [0, t_x] x \subseteq A</math> where the set 
<math display=block>a_0 + [0, t_x] x ~:=~ \left\{a_0 + t x : t \in [0, t_x]\right\}</math> 
is the line segment (or closed interval) starting at <math>a_0</math> and ending at <math>a_0 + t_x x;</math> 
this line segment is a subset of <math>a_0 + [0, \infty) x,</math> which is the {{em|ray}} emanating from <math>a_0</math> in the direction of <math>x</math> (that is, parallel to/a translation of <math>[0, \infty) x</math>). 
Thus geometrically, an interior point of a subset <math>A</math> is a point <math>a_0 \in A</math> with the property that in every possible direction (vector) <math>x \neq 0,</math> <math>A</math> contains some (non-degenerate) line segment starting at <math>a_0</math> and heading in that direction (i.e. a subset of the ray <math>a_0 + [0, \infty) x</math>). 
The algebraic interior of <math>A</math> (with respect to <math>X</math>) 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 set|radial]] points of the set.<ref name="coherent">{{cite web|title=Coherent Risk Measures, Valuation Bounds, and (<math>\mu,\rho</math>)-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}}</ref>

If <math>M</math> is a linear subspace of <math>X</math> and <math>A \subseteq X</math> then this definition can be generalized to the ''algebraic interior of <math>A</math> with respect to <math>M</math>'' is:{{sfn|Zălinescu|2002|p=2}}
<math display=block>\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\}.</math>
where <math>\operatorname{aint}_M A \subseteq A</math> always holds and if <math>\operatorname{aint}_M A \neq \varnothing</math> then <math>M \subseteq \operatorname{aff} (A - A),</math> where <math>\operatorname{aff} (A - A)</math> is the [[affine hull]] of <math>A - A</math> (which is equal to <math>\operatorname{span}(A - A)</math>).

'''Algebraic closure'''

A point <math>x \in X</math> is said to be {{em|{{visible anchor|linearly accessible}}}} from a subset <math>A \subseteq X</math> if there exists some <math>a \in A</math> such that the line segment <math>[a, x) := a + [0, 1) (x-a)</math> is contained in <math>A.</math>{{sfn|Narici|Beckenstein|2011|p=109}} 
The [[algebraic closure (convex analysis)|algebraic closure]] of <math>A</math> with respect to <math>X</math>, denoted by <math>\operatorname{acl}_X A,</math> consists of (<math>A</math> and) all points in <math>X</math> that are linearly accessible from <math>A.</math>{{sfn|Narici|Beckenstein|2011|p=109}}

==Algebraic Interior (Core)==

In the special case where <math>M := X,</math> the set <math>\operatorname{aint}_X A</math> is called the ''{{visible anchor|algebraic interior}}'' or ''{{visible anchor|core}} of <math>A</math>'' and it is denoted by <math>A^i</math> or <math>\operatorname{core} A.</math> 
Formally, if <math>X</math> is a vector space then the algebraic interior of <math>A \subseteq X</math> is<ref>{{cite book|author=Nikolaĭ Kapitonovich Nikolʹskiĭ|title=Functional analysis I: linear functional analysis|date=1992|publisher=Springer|isbn=978-3-540-50584-6}}</ref> 
<math display=block>\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\}.</math>

We call ''A'' ''algebraically open'' in ''X'' if <math>A = \operatorname{aint}_X A</math>

If <math>A</math> 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]]):

<math display=block>{}^{ic} A := 
\begin{cases}
{}^i A & \text{ if } \operatorname{aff} A \text{ is a closed set,} \\
\varnothing & \text{ otherwise}
\end{cases}
</math>

<math display=block>{}^{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}
</math>

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

===Examples===

If <math>A = \{x \in \R^2: x_2 \geq x_1^2 \text{ or } x_2 \leq 0\} \subseteq \R^2</math> then <math>0 \in \operatorname{core}(A),</math> but <math>0 \not\in \operatorname{int}(A)</math> and <math>0 \not\in \operatorname{core}(\operatorname{core}(A)).</math>

===Properties of core===

Suppose <math>A, B \subseteq X.</math> 
* In general, <math>\operatorname{core} A \neq \operatorname{core}(\operatorname{core} A).</math> But if <math>A</math> is a [[convex set]] then:
** <math>\operatorname{core} A = \operatorname{core}(\operatorname{core} A),</math> and
** for all <math>x_0 \in \operatorname{core} A, y \in A, 0 < \lambda \leq 1</math> then <math>\lambda x_0 + (1 - \lambda)y \in \operatorname{core} A.</math>
* <math>A</math> is an [[Absorbing set|absorbing subset]] of a real vector space if and only if <math>0 \in \operatorname{core}(A).</math><ref name="coherent" />
* <math>A + \operatorname{core} B \subseteq \operatorname{core}(A + B)</math>{{sfn|Zălinescu|2002|pp=2–3}}
* <math>A + \operatorname{core} B = \operatorname{core}(A + B)</math> if <math>B = \operatorname{core}B.</math>{{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 <math>C</math> is convex, <math>c \in \operatorname{core} C,</math> and <math>b \in \operatorname{acl}_X C</math> then the line segment <math>[c, b) := c + [0, 1) b</math> is contained in <math>\operatorname{core} C.</math>{{sfn|Narici|Beckenstein|2011|p=109}}

===Relation to topological interior===

Let <math>X</math> be a [[topological vector space]], <math>\operatorname{int}</math> denote the interior operator, and <math>A \subseteq X</math> then: 
* <math>\operatorname{int}A \subseteq \operatorname{core}A</math>
* If <math>A</math> is nonempty convex and <math>X</math> is finite-dimensional, then <math>\operatorname{int} A = \operatorname{core} A.</math>{{sfn|Aliprantis|Border|2006|pp=199–200}}
* If <math>A</math> is convex with non-empty interior, then <math>\operatorname{int}A = \operatorname{core} A.</math><ref name="kantorovitz">{{cite book|last=Kantorovitz|first=Shmuel|title=Introduction to Modern Analysis|publisher=[[Oxford University Press]]|date=2003|isbn=9780198526568|page=134}}</ref>
* If <math>A</math> is a closed convex set and <math>X</math> is a [[complete metric space]], then <math>\operatorname{int} A = \operatorname{core} A.</math><ref>{{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.&nbsp;56|url=https://books.google.com/books?id=ET70F9HgIpIC&pg=PA56}}.</ref>

==Relative algebraic interior==

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

==Relative interior==

If <math>A</math> is a subset of a topological vector space <math>X</math> then the ''relative interior'' of <math>A</math> is the set
<math display=block>\operatorname{rint} A := \operatorname{int}_{\operatorname{aff} A} A.</math> 
That is, it is the topological interior of A in <math>\operatorname{aff} A,</math> which is the smallest affine linear subspace of <math>X</math> containing <math>A.</math> The following set is also useful:
<math display=block>\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}
</math>

==Quasi relative interior==

If <math>A</math> is a subset of a topological vector space <math>X</math> then the ''quasi relative interior'' of <math>A</math> is the set
<math display=block>\operatorname{qri} A := \left\{ a \in A : \overline{\operatorname{cone}} (A - a) \text{ is a linear subspace of } X \right\}.</math>

In a [[Hausdorff space|Hausdorff]] finite dimensional topological vector space, <math>\operatorname{qri} A = {}^i A = {}^{ic} A = {}^{ib} A.</math>

==See also==

* [[Algebraic closure (convex analysis)]]
* {{annotated link|Bounding point}}
* {{annotated link|Interior (topology)}}
* {{annotated link|Order unit}}
* {{annotated link|Quasi-relative interior}}
* {{annotated link|Radial set}}
* {{annotated link|Relative interior}}
* {{annotated link|Ursescu theorem}}

==References ==

{{reflist|group=note}}
{{reflist|group=proof}}
{{reflist}}

===Bibliography===
{{sfn whitelist|CITEREFZălinescu2002}}
* {{Aliprantis Border Infinite Dimensional Analysis A Hitchhiker's Guide Third Edition}} <!--{{sfn|Aliprantis|Border|2006|p=}}-->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!--{{sfn|Narici|Beckenstein|2011|p=}}-->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!--{{sfn|Schaefer|Wolff|1999|p=}}-->
* {{Schechter Handbook of Analysis and Its Foundations}} <!--{{sfn|Schechter|1996|p=}}-->
* {{Zălinescu Convex Analysis in General Vector Spaces 2002}} <!--{{sfn|Zălinescu|2002|p=}}-->

{{Functional analysis}}
{{Topological vector spaces}}
{{Convex analysis and variational analysis}}

[[Category:Convex analysis]]
[[Category:Functional analysis]]
[[Category:Mathematical analysis]]
[[Category:Topology]]