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.

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 ${\displaystyle A}$¹² and ${\displaystyle A}$ is said to be radial at ${\displaystyle a_{0}}$ if for every ${\displaystyle x\in X}$ there exists a real number ${\displaystyle t_{x}>0}$ such that for every ${\displaystyle t\in [0,t_{x}],}$ ${\displaystyle a_{0}+tx\in A.}$ This last condition can also be written as ${\displaystyle a_{0}+[0,t_{x}]x\subseteq A}$ where the set $${\displaystyle a_{0}+[0,t_{x}]x~:=~\left\{a_{0}+tx:t\in [0,t_{x}]\right\}}$$ 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 $${\displaystyle \operatorname {aint} _{M}A:=\left\{a\in X:{\text{ for all }}m\in M,{\text{ there exists some }}t_{m}>0{\text{ such that }}a+\left[0,t_{m}\right]\cdot m\subseteq A\right\}.}$$ 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 ${\displaystyle [a,x):=a+[0,1)(x-a)}$ 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

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 ${\displaystyle A\subseteq X}$ is⁶ $${\displaystyle \operatorname {aint} _{X}A:=\operatorname {core} (A):=\left\{a\in A:{\text{ for all }}x\in X,{\text{ there exists some }}t_{x}>0,{\text{ such that for all }}t\in \left[0,t_{x}\right],a+tx\in A\right\}.}$$

We call A algebraically open in X if ${\displaystyle A=\operatorname {aint} _{X}A}$

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.

If ${\displaystyle A=\{x\in \mathbb {R} ^{2}:x_{2}\geq x_{1}^{2}{\text{ or }}x_{2}\leq 0\}\subseteq \mathbb {R} ^{2}}$ then ${\displaystyle 0\in \operatorname {core} (A),}$ but ${\displaystyle 0\not \in \operatorname {int} (A)}$ and ${\displaystyle 0\not \in \operatorname {core} (\operatorname {core} (A)).}$

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 ${\displaystyle x_{0}\in \operatorname {core} A,y\in A,0<\lambda \leq 1}$ 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

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

If ${\displaystyle M=\operatorname {aff} (A-A)}$ then the set ${\displaystyle \operatorname {aint} _{M}A}$ is denoted by ${\displaystyle {}^{i}A:=\operatorname {aint} _{\operatorname {aff} (A-A)}A}$ 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 ${\displaystyle a\in {}^{i}A}$ (where ${\displaystyle \operatorname {aff} A=X}$ if and only if ${\displaystyle \operatorname {aff} (A-A)=X}$).

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: $${\displaystyle \operatorname {ri} A:={\begin{cases}\operatorname {rint} A&{\text{ if }}\operatorname {aff} A{\text{ is a closed subspace of }}X{\text{,}}\\\varnothing &{\text{ otherwise}}\end{cases}}}$$

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, ${\displaystyle \operatorname {qri} A={}^{i}A={}^{ic}A={}^{ib}A.}$