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. 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 is a linear subspace of X and $A\subseteq X$ then the algebraic interior of $A$ with respect to M is:[4]

\operatorname {aint} _{M}A:=\left\{a\in X:\forall m\in M,\exists t_{m}>0{\text{ s.t. }}a+[0,t_{m}]\cdot m\subseteq A\right\}.

where it is clear that $\operatorname {aint} _{M}A\subseteq A$ and if $\operatorname {aint} _{M}A\neq \emptyset$ 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 Interior (Core)

The set $\operatorname {aint} _{X}A$ is called the algebraic interior of A or the 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

\operatorname {aint} _{X}A:=\operatorname {core} (A):=\left\{a\in A:\forall x\in X,\exists t_{x}>0,\forall t\in [0,t_{x}],a+tx\in A\right\}.

[5]

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,}}\\\emptyset &{\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{,}}\\\emptyset &{\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's possible to have ${}^{ic}A=\emptyset$ while ${}^{ib}A$ is not empty.

Example

If $A=\{x\in \mathbb {R} ^{2}:x_{2}\geq x_{1}^{2}{\text{ or }}x_{2}\leq 0\}\subseteq \mathbb {R} ^{2}$ 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

If $A,B\subset X$ then:

In general, $\operatorname {core} (A)\neq \operatorname {core} (\operatorname {core} (A))$ .
If $A$ $\text{[math]}$ 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 absorbing if and only if $0\in \operatorname {core} (A)$ .[1]
$A+\operatorname {core} B\subset \operatorname {core} (A+B)$ [6]
$A+\operatorname {core} B=\operatorname {core} (A+B)$ if $B=\operatorname {core} B$ [6]

Relation to interior

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

$\operatorname {int} A\subseteq \operatorname {core} A$
If $A$ is nonempty convex and $X$ is finite-dimensional, then $\operatorname {int} A=\operatorname {core} A$ [2]
If $A$ is convex with non-empty interior, then $\operatorname {int} A=\operatorname {core} A$ [7]
If $A$ is a closed convex set and $X$ is a complete metric space, then $\operatorname {int} A=\operatorname {core} A$ [8]

Relative algebraic interior

If $M=\operatorname {aff} (A-A)$ then the set $\operatorname {aint} _{M}A$ is denoted by ${}^{i}A:=\operatorname {aint} _{\operatorname {aff} (A-A)}A$ and it is called the relative algebraic interior of $A$ .[6] This name stems from the fact that $a\in A^{i}$ if and only if $\operatorname {aff} A=X$ and $a\in {}^{i}A$ (where $\operatorname {aff} A=X$ if and only if $\operatorname {aff} \left(A-A\right)=X$ ).

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{,}}\\\emptyset &{\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

Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ( $\mu ,\rho$ )-Portfolio Optimization". Cite journal requires |journal= (help)
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.
Zalinescu 2002, p. 2.
Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.
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.

Zalinescu, C. (2002). Convex Analysis in General Vector Spaces. World Scientific. ISBN 978-981-238-067-8.

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[coherent-1] Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ( $\mu ,\rho$ )-Portfolio Optimization". Cite journal requires |journal= (help)

[aliprantis+border-2] 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.

[cook-3] John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (pdf). Retrieved November 14, 2012.

[FOOTNOTEZalinescu20022-4] Zalinescu 2002, p. 2.

[5] Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.

[Zalinescu-6] 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.

[kantorovitz-7] Shmuel Kantorovitz (2003). Introduction to Modern Analysis. Oxford University Press. p. 134. ISBN 9780198526568.

[8] Bonnans, J. Frederic; Shapiro, Alexander (2000), Perturbation Analysis of Optimization Problems, Springer series in operations research, Springer, Remark 2.73, p. 56, ISBN 9780387987057.

Topological vector spaces (TVSs)
Basic concepts	Banach space Continuous linear operator Functionals Hilbert space Linear operators Locally convex space Homomorphism Topological vector space Vector space
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Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
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