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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]]. It is the subset of points contained in a given set that it is [[absorbing set|absorbing]] with respect to, i.e. the [[radial set|radial]] points of the set.<ref name="coherent">{{cite journal|title=Coherent Risk Measures, Valuation Bounds, and (<math>\mu,\rho</math>)-Portfolio Optimization|first1=Stefan|last1=Jaschke|first2=Uwe|last2=Kuchler|year=2000}}</ref> The elements of the algebraic interior are often referred to as '''internal points'''.<ref name="aliprantis+border">{{cite book|last=Aliprantis|first=C.D.|last2=Border|first2=K.C.|title=Infinite Dimensional Analysis: A Hitchhiker's Guide|edition=3|publisher=Springer|year=2007|isbn=978-3-540-32696-0|doi=10.1007/3-540-29587-9|pages=199–200}}</ref><ref name="cook">{{cite web|url=http://www.johndcook.com/SeparationOfConvexSets.pdf | accessdate=November 14, 2012 |format=pdf |title=Separation of Convex Sets in Linear Topological Spaces |author=John Cook |date=May 21, 1988}}</ref> | |||
Formally, if <math>X</math> is a [[linear space]] then the algebraic interior of <math>A \subseteq X</math> is | |||
: <math>\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\}.</math><ref>{{cite book|author=Nikolaĭ Kapitonovich Nikolʹskiĭ|title=Functional analysis I: linear functional analysis|year=1992|publisher=Springer|isbn=978-3-540-50584-6}}</ref> | |||
Note that 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>. In fact, if <math>A</math> is a convex set then if <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>. | |||
== Example == | |||
If <math>A \subset \mathbb{R}^2</math> such that <math>A = \{x \in \mathbb{R}^2: x_2 \geq x_1^2 \text{ or } x_2 \leq 0\}</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 == | |||
Let <math>A,B \subset X</math> then: | |||
* <math>A</math> is [[Absorbing set|absorbing]] if and only if <math>0 \in \operatorname{core}(A)</math>.<ref name="coherent" /> | |||
* <math>A + \operatorname{core}B \subset \operatorname{core}(A + B)</math><ref name="Zalinescu">{{cite book|last=Zălinescu|first=C.|title=Convex analysis in general vector spaces|publisher=World Scientific Publishing Co., Inc|location= River Edge, NJ, |year= 2002|pages=2–3|isbn=981-238-067-1|mr=1921556}}</ref> | |||
* <math>A + \operatorname{core}B = \operatorname{core}(A + B)</math> if <math>B = \operatorname{core}B</math><ref name="Zalinescu" /> | |||
=== Relation to interior === | |||
Let <math>X</math> be a [[topological vector space]], <math>\operatorname{int}</math> denote the interior operator, and <math>A \subset 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><ref name="aliprantis+border" /> | |||
* If <math>A</math> is convex with non-empty interior, then <math>\operatorname{int}A = \operatorname{core}A</math><ref name="kantorovitz">{{cite book|title=Introduction to Modern Analysis |author=Shmuel Kantorovitz |publisher=Oxford University Press |year=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|title=Perturbation Analysis of Optimization Problems|series=Springer series in operations research|first1=J. Frederic|last1=Bonnans|first2=Alexander|last2=Shapiro|publisher=Springer|year=2000|isbn=9780387987057|at=Remark 2.73, p. 56|url=http://books.google.com/books?id=ET70F9HgIpIC&pg=PA56}}.</ref> | |||
== See also == | |||
* [[Interior (topology)]] | |||
* [[Relative interior]] | |||
* [[Quasi-relative interior]] | |||
* [[Order unit]] | |||
* [[Bounding point]] | |||
== References == | |||
{{Reflist}} | |||
{{Functional Analysis}} | |||
[[Category:Topology]] | |||
[[Category:Mathematical analysis]] | |||
[[Category:Functional analysis]] | |||
Revision as of 14:17, 20 August 2013
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 that it is absorbing with respect to, i.e. the radial points of the set.[1] The elements of the algebraic interior are often referred to as internal points.[2][3]
Formally, if is a linear space then the algebraic interior of is
![{\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\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ad783b2516aa4afe53659ee2105cc18ac096401)
Note that in general , but if is a convex set then . In fact, if is a convex set then if then .
Example
If such that then , but and .
Properties
Let then:
Relation to interior
Let be a topological vector space, denote the interior operator, and then:
- If is nonempty convex and is finite dimensional, then [2]
- If is convex with non-empty interior, then [6]
- If is a closed convex set and is a complete metric space, then [7]
See also
References
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