Algebraic Interior (Core)

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

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

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 $\Omega$.

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

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 $\operatorname{core}(L)=L$.