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 internal point of $A$ and $A$ is said to be radial at $a_{0}$ if for every $x\in X$ there exists a real number $t_{x}>0$ such that for every $t\in [0,t_{x}],$ $a_{0}+tx\in A.$ This last condition can also be written as $a_{0}+[0,t_{x}]x\subseteq A$ where the set $$a_{0}+[0,t_{x}]x~:=~\left\{a_{0}+tx:t\in [0,t_{x}]\right\}$$ 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 ray 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 linearly accessible 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 algebraic closure of $A$ with respect to $X$, 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 algebraic interior or core 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 not empty.

Examples

If $A=\{x\in \mathbb {R} ^{2}:x_{2}\geq x_{1}^{2}{\text{ or }}x_{2}\leq 0\}\subseteq \mathbb {R} ^{2}$ 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 ${}^{i}A:=\operatorname {aint} _{\operatorname {aff} (A-A)}A$ 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 $a\in {}^{i}A$ (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, $\operatorname {qri} A={}^{i}A={}^{ic}A={}^{ib}A.$

See also

• Bounding point – Mathematical concept related to subsets of vector spaces
• Interior (topology) – Largest open subset of some given set
• Order unit – Element of an ordered vector space
• Quasi-relative interior – Generalization of algebraic interior
• Radial set
• Relative interior – Generalization of topological interior
• Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem

Reference

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