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 ${\displaystyle A\subseteq X}$ then the algebraic interior of ${\displaystyle A}$ with respect to M is:[4]

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

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

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

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

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

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

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

Example

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

If ${\displaystyle A,B\subset X}$ then:

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

Relation to interior

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

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

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 ${\displaystyle A}$.[6] 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 ${\displaystyle \operatorname {aff} \left(A-A\right)=X}$).

Relative interior

If A is a subset of a topological vector space X then the relative interior of A is the set

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

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

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

Quasi relative interior

If A is a subset of a topological vector space X then the quasi relative interior of A is the set

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

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

See also

• Bounding point
• Interior (topology)
• Quasi-relative interior
• Relative interior
• Order unit
• Ursescu theorem

References

1. Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and (${\displaystyle \mu ,\rho }$)-Portfolio Optimization". Cite journal requires |journal= (help)
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.
3. John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces" (pdf). Retrieved November 14, 2012.
4. Zalinescu 2002, p. 2.
5. Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.
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.
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.

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