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.

Definition
Assume that $$A$$ is a subset of a vector space $$X.$$ The algebraic interior (or radial kernel) of $$A$$ with respect to $$X$$ is the set of all points at which $$A$$ is a radial set. A point $$a_0 \in A$$ is called an of $$A$$ and $$A$$ is said to be  if for every $$x \in X$$ there exists a real number $$t_x > 0$$ such that for every $$t \in [0, t_x],$$ $$a_0 + t x \in A.$$ This last condition can also be written as $$a_0 + [0, t_x] x \subseteq A$$ where the set $$a_0 + [0, t_x] x ~:=~ \left\{a_0 + t x : t \in [0, t_x]\right\}$$ is the line segment (or closed interval) starting at $$a_0$$ and ending at $$a_0 + t_x x;$$ this line segment is a subset of $$a_0 + [0, \infty) x,$$ which is the emanating from $$a_0$$ in the direction of $$x$$ (that is, parallel to/a translation of $$[0, \infty) x$$). Thus geometrically, an interior point of a subset $$A$$ is a point $$a_0 \in A$$ with the property that in every possible direction (vector) $$x \neq 0,$$ $$A$$ contains some (non-degenerate) line segment starting at $$a_0$$ and heading in that direction (i.e. a subset of the ray $$a_0 + [0, \infty) x$$). The algebraic interior of $$A$$ (with respect to $$X$$) is the set of all such points. That is to say, it is the subset of points contained in a given set with respect to which it is radial points of the set.

If $$M$$ is a linear subspace of $$X$$ and $$A \subseteq X$$ then this definition can be generalized to the algebraic interior of $$A$$ with respect to $$M$$ is: $$\operatorname{aint}_M A := \left\{ a \in X : \text{ for all } m \in M, \text{ there exists some } t_m > 0 \text{ such that } a + \left[0, t_m\right] \cdot m \subseteq A \right\}.$$ where $$\operatorname{aint}_M A \subseteq A$$ always holds and if $$\operatorname{aint}_M A \neq \varnothing$$ then $$M \subseteq \operatorname{aff} (A - A),$$ where $$\operatorname{aff} (A - A)$$ is the affine hull of $$A - A$$ (which is equal to $$\operatorname{span}(A - A)$$).

Algebraic closure

A point $$x \in X$$ is said to be from a subset $$A \subseteq X$$ if there exists some $$a \in A$$ such that the line segment $$[a, x) := a + [0, 1) x$$ is contained in $$A.$$ The, denoted by $$\operatorname{acl}_X A,$$ consists of $$A$$ and all points in $$X$$ that are linearly accessible from $$A.$$

Algebraic Interior (Core)
In the special case where $$M := X,$$ the set $$\operatorname{aint}_X A$$ is called the  or  of $$A$$ and it is denoted by $$A^i$$ or $$\operatorname{core} A.$$ Formally, if $$X$$ is a vector space then the algebraic interior of $$A \subseteq X$$ is $$\operatorname{aint}_X A := \operatorname{core}(A) := \left\{ a \in A : \text{ for all } x \in X, \text{ there exists some } t_x > 0, \text{ such that for all } t \in \left[0, t_x\right], a + tx \in A \right\}.$$

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):

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

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

If $$X$$ is a Fréchet space, $$A$$ is convex, and $$\operatorname{aff} A$$ is closed in $$X$$ then $${}^{ic} A = {}^{ib} A$$ but in general it is possible to have $${}^{ic} A = \varnothing$$ while $${}^{ib} A$$ is empty.

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

Properties of core
Suppose $$A, B \subseteq X.$$
 * In general, $$\operatorname{core} A \neq \operatorname{core}(\operatorname{core} A).$$ But if $$A$$ is a convex set then:
 * $$\operatorname{core} A = \operatorname{core}(\operatorname{core} A),$$ and
 * for all $$x_0 \in \operatorname{core} A, y \in A, 0 < \lambda \leq 1$$ then $$\lambda x_0 + (1 - \lambda)y \in \operatorname{core} A.$$
 * $$A$$ is an absorbing subset of a real vector space if and only if $$0 \in \operatorname{core}(A).$$
 * $$A + \operatorname{core} B \subseteq \operatorname{core}(A + B)$$
 * $$A + \operatorname{core} B = \operatorname{core}(A + B)$$ if $$B = \operatorname{core}B.$$

Both the core and the algebraic closure of a convex set are again convex. If $$C$$ is convex, $$c \in \operatorname{core} C,$$ and $$b \in \operatorname{acl}_X C$$ then the line segment $$[c, b) := c + [0, 1) b$$ is contained in $$\operatorname{core} C.$$

Relation to topological interior
Let $$X$$ be a topological vector space, $$\operatorname{int}$$ denote the interior operator, and $$A \subseteq X$$ 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 $$\operatorname{int} A = \operatorname{core} A.$$

Relative algebraic interior
If $$M = \operatorname{aff} (A - A)$$ then the set $$\operatorname{aint}_M A$$ is denoted by $${}^iA := \operatorname{aint}_{\operatorname{aff} (A - A)} A$$ and it is called the relative algebraic interior of $$A.$$ This name stems from the fact that $$a \in A^i$$ if and only if $$\operatorname{aff} A = X$$ and $$a \in {}^iA$$ (where $$\operatorname{aff} A = X$$ if and only if $$\operatorname{aff} (A - A) = X$$).

Relative interior
If $$A$$ is a subset of a topological vector space $$X$$ then the relative interior of $$A$$ is the set $$\operatorname{rint} A := \operatorname{int}_{\operatorname{aff} A} A.$$ That is, it is the topological interior of A in $$\operatorname{aff} A,$$ which is the smallest affine linear subspace of $$X$$ containing $$A.$$ The following set is also useful: $$\operatorname{ri} A := \begin{cases} \operatorname{rint} A & \text{ if } \operatorname{aff} A \text{ is a closed subspace of } X \text{,} \\ \varnothing & \text{ otherwise} \end{cases} $$

Quasi relative interior
If $$A$$ is a subset of a topological vector space $$X$$ then the quasi relative interior of $$A$$ is the set $$\operatorname{qri} A := \left\{ a \in A : \overline{\operatorname{cone}} (A - a) \text{ is a linear subspace of } X \right\}.$$

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