# Algebraic interior

Short description: Generalization of topological 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 $\displaystyle{ A }$ is a subset of a vector space $\displaystyle{ X. }$ The algebraic interior (or radial kernel) of $\displaystyle{ A }$ with respect to $\displaystyle{ X }$ is the set of all points at which $\displaystyle{ A }$ is a radial set. A point $\displaystyle{ a_0 \in A }$ is called an internal point of $\displaystyle{ A }$[1][2] and $\displaystyle{ A }$ is said to be radial at $\displaystyle{ a_0 }$ if for every $\displaystyle{ x \in X }$ there exists a real number $\displaystyle{ t_x \gt 0 }$ such that for every $\displaystyle{ t \in [0, t_x], }$ $\displaystyle{ a_0 + t x \in A. }$ This last condition can also be written as $\displaystyle{ a_0 + [0, t_x] x \subseteq A }$ where the set $\displaystyle{ a_0 + [0, t_x] x ~:=~ \left\{a_0 + t x : t \in [0, t_x]\right\} }$ is the line segment (or closed interval) starting at $\displaystyle{ a_0 }$ and ending at $\displaystyle{ a_0 + t_x x; }$ this line segment is a subset of $\displaystyle{ a_0 + [0, \infty) x, }$ which is the ray emanating from $\displaystyle{ a_0 }$ in the direction of $\displaystyle{ x }$ (that is, parallel to/a translation of $\displaystyle{ [0, \infty) x }$). Thus geometrically, an interior point of a subset $\displaystyle{ A }$ is a point $\displaystyle{ a_0 \in A }$ with the property that in every possible direction (vector) $\displaystyle{ x \neq 0, }$ $\displaystyle{ A }$ contains some (non-degenerate) line segment starting at $\displaystyle{ a_0 }$ and heading in that direction (i.e. a subset of the ray $\displaystyle{ a_0 + [0, \infty) x }$). The algebraic interior of $\displaystyle{ A }$ (with respect to $\displaystyle{ X }$) 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.[3]

If $\displaystyle{ M }$ is a linear subspace of $\displaystyle{ X }$ and $\displaystyle{ A \subseteq X }$ then this definition can be generalized to the algebraic interior of $\displaystyle{ A }$ with respect to $\displaystyle{ M }$ is:[4] $\displaystyle{ \operatorname{aint}_M A := \left\{ a \in X : \text{ for all } m \in M, \text{ there exists some } t_m \gt 0 \text{ such that } a + \left[0, t_m\right] \cdot m \subseteq A \right\}. }$ where $\displaystyle{ \operatorname{aint}_M A \subseteq A }$ always holds and if $\displaystyle{ \operatorname{aint}_M A \neq \varnothing }$ then $\displaystyle{ M \subseteq \operatorname{aff} (A - A), }$ where $\displaystyle{ \operatorname{aff} (A - A) }$ is the affine hull of $\displaystyle{ A - A }$ (which is equal to $\displaystyle{ \operatorname{span}(A - A) }$).

Algebraic closure

A point $\displaystyle{ x \in X }$ is said to be linearly accessible from a subset $\displaystyle{ A \subseteq X }$ if there exists some $\displaystyle{ a \in A }$ such that the line segment $\displaystyle{ [a, x) := a + [0, 1) x }$ is contained in $\displaystyle{ A. }$[5] The algebraic closure of $\displaystyle{ A }$ with respect to $\displaystyle{ X }$, denoted by $\displaystyle{ \operatorname{acl}_X A, }$ consists of $\displaystyle{ A }$ and all points in $\displaystyle{ X }$ that are linearly accessible from $\displaystyle{ A. }$[5]

## Algebraic Interior (Core)

In the special case where $\displaystyle{ M := X, }$ the set $\displaystyle{ \operatorname{aint}_X A }$ is called the algebraic interior or core of $\displaystyle{ A }$ and it is denoted by $\displaystyle{ A^i }$ or $\displaystyle{ \operatorname{core} A. }$ Formally, if $\displaystyle{ X }$ is a vector space then the algebraic interior of $\displaystyle{ A \subseteq X }$ is[6] $\displaystyle{ \operatorname{aint}_X A := \operatorname{core}(A) := \left\{ a \in A : \text{ for all } x \in X, \text{ there exists some } t_x \gt 0, \text{ such that for all } t \in \left[0, t_x\right], a + tx \in A \right\}. }$

If $\displaystyle{ A }$ 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):

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

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

If $\displaystyle{ X }$ is a Fréchet space, $\displaystyle{ A }$ is convex, and $\displaystyle{ \operatorname{aff} A }$ is closed in $\displaystyle{ X }$ then $\displaystyle{ {}^{ic} A = {}^{ib} A }$ but in general it is possible to have $\displaystyle{ {}^{ic} A = \varnothing }$ while $\displaystyle{ {}^{ib} A }$ is not empty.

### Examples

If $\displaystyle{ A = \{x \in \R^2: x_2 \geq x_1^2 \text{ or } x_2 \leq 0\} \subseteq \R^2 }$ 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 of core

Suppose $\displaystyle{ A, B \subseteq X. }$

• In general, $\displaystyle{ \operatorname{core} A \neq \operatorname{core}(\operatorname{core} A). }$ But if $\displaystyle{ A }$ is a convex set then:
• $\displaystyle{ \operatorname{core} A = \operatorname{core}(\operatorname{core} A), }$ and
• for all $\displaystyle{ x_0 \in \operatorname{core} A, y \in A, 0 \lt \lambda \leq 1 }$ then $\displaystyle{ \lambda x_0 + (1 - \lambda)y \in \operatorname{core} A. }$
• $\displaystyle{ A }$ is an absorbing subset of a real vector space if and only if $\displaystyle{ 0 \in \operatorname{core}(A). }$[3]
• $\displaystyle{ A + \operatorname{core} B \subseteq \operatorname{core}(A + B) }$[7]
• $\displaystyle{ A + \operatorname{core} B = \operatorname{core}(A + B) }$ if $\displaystyle{ B = \operatorname{core}B. }$[7]

Both the core and the algebraic closure of a convex set are again convex.[5] If $\displaystyle{ C }$ is convex, $\displaystyle{ c \in \operatorname{core} C, }$ and $\displaystyle{ b \in \operatorname{acl}_X C }$ then the line segment $\displaystyle{ [c, b) := c + [0, 1) b }$ is contained in $\displaystyle{ \operatorname{core} C. }$[5]

### Relation to topological interior

Let $\displaystyle{ X }$ be a topological vector space, $\displaystyle{ \operatorname{int} }$ denote the interior operator, and $\displaystyle{ A \subseteq X }$ then:

• $\displaystyle{ \operatorname{int}A \subseteq \operatorname{core}A }$
• If $\displaystyle{ A }$ is nonempty convex and $\displaystyle{ X }$ is finite-dimensional, then $\displaystyle{ \operatorname{int} A = \operatorname{core} A. }$[1]
• If $\displaystyle{ A }$ is convex with non-empty interior, then $\displaystyle{ \operatorname{int}A = \operatorname{core} A. }$[8]
• If $\displaystyle{ A }$ is a closed convex set and $\displaystyle{ X }$ is a complete metric space, then $\displaystyle{ \operatorname{int} A = \operatorname{core} A. }$[9]

## Relative algebraic interior

If $\displaystyle{ M = \operatorname{aff} (A - A) }$ then the set $\displaystyle{ \operatorname{aint}_M A }$ is denoted by $\displaystyle{ {}^iA := \operatorname{aint}_{\operatorname{aff} (A - A)} A }$ and it is called the relative algebraic interior of $\displaystyle{ A. }$[7] This name stems from the fact that $\displaystyle{ a \in A^i }$ if and only if $\displaystyle{ \operatorname{aff} A = X }$ and $\displaystyle{ a \in {}^iA }$ (where $\displaystyle{ \operatorname{aff} A = X }$ if and only if $\displaystyle{ \operatorname{aff} (A - A) = X }$).

## Relative interior

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

## Quasi relative interior

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

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

## References

1. Aliprantis & Border 2006, pp. 199–200.
2. John Cook (May 21, 1988). "Separation of Convex Sets in Linear Topological Spaces".
3. Jaschke, Stefan; Kuchler, Uwe (2000). Coherent Risk Measures, Valuation Bounds, and ($\displaystyle{ \mu,\rho }$)-Portfolio Optimization.
4. Zălinescu 2002, p. 2.
5. Narici & Beckenstein 2011, p. 109.
6. Nikolaĭ Kapitonovich Nikolʹskiĭ (1992). Functional analysis I: linear functional analysis. Springer. ISBN 978-3-540-50584-6.
7. Zălinescu 2002, pp. 2–3.
8. Kantorovitz, Shmuel (2003). Introduction to Modern Analysis. Oxford University Press. p. 134. ISBN 9780198526568.
9. Bonnans, J. Frederic; Shapiro, Alexander (2000), Perturbation Analysis of Optimization Problems, Springer series in operations research, Springer, Remark 2.73, p. 56, ISBN 9780387987057 .