Algebraic Interior (Core) - Imported Knowlpedia Content

Let $X$ be a real vector space and let $\Omega\subset X$ .

The algebraic interior (or core) of $\Omega$ is

\operatorname{core}(\Omega):=\Big\{x\in\Omega \ \Big|\ \forall v\in X,\ \exists \delta>0\ \text{s.t.}\ x+tv\in\Omega\ \text{for all }|t|<\delta\Big\}.

Equivalent characterizations

Equivalently, $x\in\operatorname{core}(\Omega)$ iff for every direction $v\in X$ , one can move a small amount from $x$ in the direction $v$ and remain in $\Omega$ .

Remarks

When $X$ is a normed vector space and $\Omega$ is convex, we have

\operatorname{int}(\Omega)\subset \operatorname{core}(\Omega)\subset \Omega,

where $\operatorname{int}(\Omega)$ is the usual interior. See also linear closure for the dual notion.

Examples

If $\Omega$ is an open ball in a normed space, then $\operatorname{core}(\Omega)=\Omega$ .
If $\Omega$ is a linear subspace $L$ , then $\operatorname{core}(L)=L$ .