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 $A$ 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 $$a_0+[0,t_x]x:=\{a_0+tx:t\in [0,t_x]\}$$ 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 $a_0+[0,\mathrm{\infty })x,$ which is the ray emanating from $a_0$ in the direction of $x$ (that is, parallel to/a translation of $[0,\mathrm{\infty })x$). 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) $x\ne 0,$ $A$ contains some (non-degenerate) line segment starting at $a_0$ and heading in that direction (i.e. a subset of the ray $a_0+[0,\mathrm{\infty })x$). 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: $${\mathrm{aint}}_{M}A:=\{a\in X:\text{ for all }m\in M,\text{ there exists some }{t}_m>0\text{ such that }a+[0,t_m]\cdot m\subseteq A\}.$$ where ${\mathrm{aint}}_{M}A\subseteq A$ always holds and if ${\mathrm{aint}}_{M}A\ne \varnothing$ then $M\subseteq \mathrm{aff}(A-A),$ where $\mathrm{aff}(A-A)$ is the affine hull of $A-A$ (which is equal to $\mathrm{span}(A-A)$).

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 ${\mathrm{acl}}_{X}A,$ 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 ${\mathrm{aint}}_{X}A$ is called the algebraic interior or core of $A$ and it is denoted by $A^i$ or $\mathrm{core}A.$ Formally, if $X$ is a vector space then the algebraic interior of $A\subseteq X$ is $${\mathrm{aint}}_{X}A:=\mathrm{core}(A):=\{a\in A:\text{ for all }x\in X,\text{ there exists some }{t}_x>0,\text{ such that for all }t\in [0,t_x],a+tx\in A\}.$$

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):

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

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

If $X$ is a Fréchet space, $A$ is convex, and $\mathrm{aff}A$ 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 $A=\{x\in {\mathbb{ℝ}}^2:x_2\ge x_1^2\text{ or }{x}_2\le 0\}\subseteq {\mathbb{ℝ}}^2$ then $0\in \mathrm{core}(A),$ but $0\in ̸\mathrm{int}(A)$ and $0\in ̸\mathrm{core}(\mathrm{core}(A)).$

Properties of core

Suppose $A,B\subseteq X.$

• In general, $\mathrm{core}A\ne \mathrm{core}(\mathrm{core}A).$ But if $A$ is a convex set then:
  • $\mathrm{core}A=\mathrm{core}(\mathrm{core}A),$ and
  • for all $x_0\in \mathrm{core}A,y\in A,0<\lambda \le 1$ then $\lambda x_0+(1-\lambda )y\in \mathrm{core}A.$
• $A$ is an absorbing subset of a real vector space if and only if $0\in \mathrm{core}(A).$
• $A+\mathrm{core}B\subseteq \mathrm{core}(A+B)$
• $A+\mathrm{core}B=\mathrm{core}(A+B)$ if $B=\mathrm{core}B.$

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

Relation to topological interior

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

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

Relative algebraic interior

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

Relative interior

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

Quasi relative interior

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

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

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

References

Bibliography

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• Template:Zălinescu Convex Analysis in General Vector Spaces 2002