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 ${\displaystyle A}$ is a subset of a vector space ${\displaystyle X.}$ The algebraic interior (or radial kernel) of ${\displaystyle A}$ with respect to ${\displaystyle X}$ is the set of all points at which ${\displaystyle A}$ is a radial set. A point ${\displaystyle a_{0}\in A}$ is called an internal point of ${\displaystyle A}$ ^{[1] [2]} and ${\displaystyle A}$ is said to be radial at ${\displaystyle a_{0}}$ if for every ${\displaystyle x\in X}$ there exists a real number ${\displaystyle t_{x}>0}$ such that for every ${\displaystyle t\in [0,t_{x}],}$${\displaystyle a_{0}+tx\in A.}$ This last condition can also be written as ${\displaystyle a_{0}+[0,t_{x}]x\subseteq A}$ where the set

$${\displaystyle a_{0}+[0,t_{x}]x~:=~\left\{a_{0}+tx:t\in [0,t_{x}]\right\}}$$

is the line segment (or closed interval) starting at ${\displaystyle a_{0}}$ and ending at ${\displaystyle a_{0}+t_{x}x;}$ this line segment is a subset of ${\displaystyle a_{0}+[0,\infty )x,}$ which is the ray emanating from ${\displaystyle a_{0}}$ in the direction of ${\displaystyle x}$ (that is, parallel to/a translation of ${\displaystyle [0,\infty )x}$). Thus geometrically, an interior point of a subset ${\displaystyle A}$ is a point ${\displaystyle a_{0}\in A}$ with the property that in every possible direction (vector) ${\displaystyle x\neq 0,}$${\displaystyle A}$ contains some (non-degenerate) line segment starting at ${\displaystyle a_{0}}$ and heading in that direction (i.e. a subset of the ray ${\displaystyle a_{0}+[0,\infty )x}$). The algebraic interior of ${\displaystyle A}$ (with respect to ${\displaystyle X}$) 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 ${\displaystyle M}$ is a linear subspace of ${\displaystyle X}$ and ${\displaystyle A\subseteq X}$ then this definition can be generalized to the algebraic interior of ${\displaystyle A}$ with respect to ${\displaystyle M}$ is: ^[4]

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

where ${\displaystyle \operatorname {aint} _{M}A\subseteq A}$ always holds and if ${\displaystyle \operatorname {aint} _{M}A\neq \varnothing }$ then ${\displaystyle M\subseteq \operatorname {aff} (A-A),}$ where ${\displaystyle \operatorname {aff} (A-A)}$ is the affine hull of ${\displaystyle A-A}$ (which is equal to ${\displaystyle \operatorname {span} (A-A)}$).

Algebraic closure

A point ${\displaystyle x\in X}$ is said to be linearly accessible from a subset ${\displaystyle A\subseteq X}$ if there exists some ${\displaystyle a\in A}$ such that the line segment ${\displaystyle [a,x):=a+[0,1)x}$ is contained in ${\displaystyle A.}$ ^[5] The algebraic closure of ${\displaystyle A}$ with respect to ${\displaystyle X}$, denoted by ${\displaystyle \operatorname {acl} _{X}A,}$ consists of ${\displaystyle A}$ and all points in ${\displaystyle X}$ that are linearly accessible from ${\displaystyle A.}$ ^[5]

Algebraic Interior (Core)

In the special case where ${\displaystyle M:=X,}$ the set ${\displaystyle \operatorname {aint} _{X}A}$ is called the algebraic interior or core of ${\displaystyle A}$ and it is denoted by ${\displaystyle A^{i}}$ or ${\displaystyle \operatorname {core} A.}$ Formally, if ${\displaystyle X}$ is a vector space then the algebraic interior of ${\displaystyle A\subseteq X}$ is ^[6]

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

If ${\displaystyle A}$ 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):

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

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

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

Examples

If ${\displaystyle A=\{x\in \mathbb {R} ^{2}:x_{2}\geq x_{1}^{2}{\text{ or }}x_{2}\leq 0\}\subseteq \mathbb {R} ^{2}}$ then ${\displaystyle 0\in \operatorname {core} (A),}$ but ${\displaystyle 0\not \in \operatorname {int} (A)}$ and ${\displaystyle 0\not \in \operatorname {core} (\operatorname {core} (A)).}$

Properties of core

Suppose ${\displaystyle A,B\subseteq X.}$

• In general, ${\displaystyle \operatorname {core} A\neq \operatorname {core} (\operatorname {core} A).}$ But if ${\displaystyle A}$ is a convex set then:
  • ${\displaystyle \operatorname {core} A=\operatorname {core} (\operatorname {core} A),}$ and
  • for all ${\displaystyle x_{0}\in \operatorname {core} A,y\in A,0<\lambda \leq 1}$ then ${\displaystyle \lambda x_{0}+(1-\lambda )y\in \operatorname {core} A.}$
• ${\displaystyle A}$ is an absorbing subset of a real vector space if and only if ${\displaystyle 0\in \operatorname {core} (A).}$ ^[3]
• ${\displaystyle A+\operatorname {core} B\subseteq \operatorname {core} (A+B)}$ ^[7]
• ${\displaystyle A+\operatorname {core} B=\operatorname {core} (A+B)}$ if ${\displaystyle B=\operatorname {core} B.}$ ^[7]

Both the core and the algebraic closure of a convex set are again convex. ^[5] If ${\displaystyle C}$ is convex, ${\displaystyle c\in \operatorname {core} C,}$ and ${\displaystyle b\in \operatorname {acl} _{X}C}$ then the line segment ${\displaystyle [c,b):=c+[0,1)b}$ is contained in ${\displaystyle \operatorname {core} C.}$ ^[5]

Relation to topological interior

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

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

Relative algebraic interior

If ${\displaystyle M=\operatorname {aff} (A-A)}$ then the set ${\displaystyle \operatorname {aint} _{M}A}$ is denoted by ${\displaystyle {}^{i}A:=\operatorname {aint} _{\operatorname {aff} (A-A)}A}$ and it is called the relative algebraic interior of ${\displaystyle A.}$ ^[7] This name stems from the fact that ${\displaystyle a\in A^{i}}$ if and only if ${\displaystyle \operatorname {aff} A=X}$ and ${\displaystyle a\in {}^{i}A}$ (where ${\displaystyle \operatorname {aff} A=X}$ if and only if ${\displaystyle \operatorname {aff} (A-A)=X}$).

Relative interior

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

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

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

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

Quasi relative interior

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

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

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

See also

• Bounding point  – Mathematical concept related to subsets of vector spaces
• Interior (topology)  – Largest open subset of some given set
• Order unit  – Element of an ordered vector space
• Quasi-relative interior  – Generalization of algebraic interior
• Radial set
• Relative interior  – Generalization of topological interior
• Ursescu theorem  – Generalization of closed graph, open mapping, and uniform boundedness theorem

References

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

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.