Algebraic interior

The set ${\displaystyle \operatorname {aint} _{X}A}$ is called the algebraic interior of ${\displaystyle A}$ or the 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

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

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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's possible to have ${\displaystyle {}^{ic}A=\varnothing }$ while ${\displaystyle {}^{ib}A}$ is not empty.

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 ${\displaystyle A.}$[6] 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, ${\displaystyle \operatorname {qri} A={}^{i}A={}^{ic}A={}^{ib}A.}$

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

• Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.