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. The elements of the algebraic interior are often referred to as internal points.

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

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

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 $0 \in \operatorname{core}(A)$.
• $A + \operatorname{core}B \subset \operatorname{core}(A + B)$
• $A + \operatorname{core}B = \operatorname{core}(A + B)$ if $B = \operatorname{core}B$

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 $\operatorname{int}A = \operatorname{core}A$
• If $A$ is convex with non-empty interior, then $\operatorname{int}A = \operatorname{core}A$
• If $A$ is a closed convex set and $X$ is a complete metric space, then $\operatorname{int}A = \operatorname{core}A$

See also

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