# Algebraic interior

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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. 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]

Formally, if ${\displaystyle X}$ is a linear space then the algebraic interior of ${\displaystyle A\subseteq X}$ is

${\displaystyle \operatorname {core} (A):=\left\{x_{0}\in A:\forall x\in X,\exists t_{x}>0,\forall t\in [0,t_{x}],x_{0}+tx\in A\right\}.}$[4]

Note that in general ${\displaystyle \operatorname {core} (A)\neq \operatorname {core} (\operatorname {core} (A))}$, but if ${\displaystyle A}$ is a convex set then ${\displaystyle \operatorname {core} (A)=\operatorname {core} (\operatorname {core} (A))}$. If ${\displaystyle A}$ is a convex set then if ${\displaystyle x_{0}\in \operatorname {core} (A),y\in A,0<\lambda \leq 1}$ then ${\displaystyle \lambda x_{0}+(1-\lambda )y\in \operatorname {core} (A)}$.

## 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

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

• ${\displaystyle A}$ is absorbing if and only if ${\displaystyle 0\in \operatorname {core} (A)}$.[1]
• ${\displaystyle A+\operatorname {core} B\subset \operatorname {core} (A+B)}$[5]
• ${\displaystyle A+\operatorname {core} B=\operatorname {core} (A+B)}$ if ${\displaystyle B=\operatorname {core} B}$[5]

### 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 ${\displaystyle \operatorname {int} A=\operatorname {core} A}$[6]
• If ${\displaystyle A}$ is a closed convex set and ${\displaystyle X}$ is a complete metric space, then ${\displaystyle \operatorname {int} A=\operatorname {core} A}$[7]

## References

1. ^ a b Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and (${\displaystyle \mu ,\rho }$)-Portfolio Optimization".
2. ^ a b 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. ^ Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.
5. ^ a b 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.
6. ^ Shmuel Kantorovitz (2003). Introduction to Modern Analysis. Oxford University Press. p. 134. ISBN 9780198526568.
7. ^ Bonnans, J. Frederic; Shapiro, Alexander (2000), Perturbation Analysis of Optimization Problems, Springer series in operations research, Springer, Remark 2.73, p. 56, ISBN 9780387987057.