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 internal point of $A$ ^[1]^[2] and $A$ is said to be radial at $a_{0}$ if for every $x\in X$ there exists a real number $t_{x}>0$ such that for every $t\in [0,t_{x}],$ $a_{0}+tx\in A.$ This last condition can also be written as $a_{0}+[0,t_{x}]x\subseteq A$ where the set $a_{0}+[0,t_{x}]x~:=~\left\{a_{0}+tx:t\in [0,t_{x}]\right\}$ 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 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.^[3]

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:^[4] $\operatorname {aint} _{M}A:=\left\{a\in X:{\text{ for all }}m\in M,{\text{ there exists some }}t_{m}>0{\text{ such that }}a+\left[0,t_{m}\right]\cdot m\subseteq A\right\}.$ 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 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 $A.$ ^[5] 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 $A.$ ^[5]

Algebraic Interior (Core)

In the special case where $M:=X,$ the set $\operatorname {aint} _{X}A$ is called the algebraic interior or 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^[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 not empty.

Examples

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

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 $0\in \operatorname {core} (A).$ ^[3]
$A+\operatorname {core} B\subseteq \operatorname {core} (A+B)$ ^[7]
$A+\operatorname {core} B=\operatorname {core} (A+B)$ if $B=\operatorname {core} B.$ ^[7]

Both the core and the algebraic closure of a convex set are again convex.^[5] 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 $\operatorname {core} C.$ ^[5]

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 $\operatorname {int} A=\operatorname {core} A.$ ^[1]
If $A$ is convex with non-empty interior, then $\operatorname {int} A=\operatorname {core} A.$ ^[8]
If $A$ is a closed convex set and $X$ is a complete metric space, then $\operatorname {int} A=\operatorname {core} A.$ ^[9]

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 $A.$ ^[7] 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 $\operatorname {aff} (A-A)=X$ ).

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

^ ^a ^b Aliprantis & Border 2006, pp. 199–200.
^ John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (PDF). Retrieved November 14, 2012.
^ ^a ^b Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ( $\mu ,\rho$ )-Portfolio Optimization" (PDF).
^ Zălinescu 2002, p. 2.
^ ^a ^b ^c ^d Narici & Beckenstein 2011, p. 109.
^ Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.
^ ^a ^b ^c Zălinescu 2002, pp. 2–3.
^ Kantorovitz, Shmuel (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.

Bibliography

Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide (Third ed.). Berlin: Springer Science & Business Media. ISBN 978-3-540-29587-7. OCLC 262692874.
Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.

[FOOTNOTEAliprantisBorder2006199–200-1] Aliprantis & Border 2006, pp. 199–200.

[cook-2] John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (PDF). Retrieved November 14, 2012.

[coherent-3] Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ( $\mu ,\rho$ )-Portfolio Optimization" (PDF).

[FOOTNOTEZălinescu20022-4] Zălinescu 2002, p. 2.

[FOOTNOTENariciBeckenstein2011109-5] Narici & Beckenstein 2011, p. 109.

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

[FOOTNOTEZălinescu20022–3-7] Zălinescu 2002, pp. 2–3.

[kantorovitz-8] Kantorovitz, Shmuel (2003). Introduction to Modern Analysis. Oxford University Press. p. 134. ISBN 9780198526568.

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

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