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]

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

$\operatorname{core}(A) := \left\{x_0 \in A: \forall x \in X, \exists t_x > 0, \forall t \in [0,t_x], x_0 + tx \in A\right\}.$^[4]

Note that in general $\operatorname{core}(A) \neq \operatorname{core}(\operatorname{core}(A))$, but if $A$ is a convex set then $\operatorname{core}(A) = \operatorname{core}(\operatorname{core}(A))$. If $A$ is a convex set then if $x_0 \in \operatorname{core}(A), y \in A, 0 < \lambda \leq 1$ then $\lambda x_0 + (1 - \lambda)y \in \operatorname{core}(A)$.

Example

If $A \subset \mathbb{R}^2$ such that $A = \{x \in \mathbb{R}^2: x_2 \geq x_1^2 \text{ or } x_2 \leq 0\}$ then $0 \in \operatorname{core}(A)$, but $0 \not\in \operatorname{int}(A)$ and $0 \not\in \operatorname{core}(\operatorname{core}(A))$.

Properties

Let $A,B \subset X$ then:

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

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

See also

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

References

1. Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ($\mu,\rho$)-Portfolio Optimization". delete character in |title= at position 48 (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. Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.
5. 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.