# 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 that it is absorbing with respect to, 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 $\operatorname{core}(A) = \operatorname{core}(\operatorname{core}(A))$. In fact, 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]

1. ^ a b Jaschke, Stefan; Kuchler, Uwe (2000). "Coherent Risk Measures, Valuation Bounds, and ($\mu,\rho$)-Portfolio Optimization".