# relative interior

Let $S$ be a subset of the $n$-dimensional Euclidean space $\mathbb{R}^{n}$. The relative interior of $S$ is the interior of $S$ considered as a subset of its affine hull $\operatorname{Aff}(S)$, and is denoted by $\operatorname{ri}(S)$.

The difference between the interior and the relative interior of $S$ can be illustrated in the following two examples. Consider the closed unit square

 $I^{2}:=\{(x,y,0)\mid 0\leq x,y\leq 1\}$

in $\mathbb{R}^{3}$. Its interior is $\varnothing$, the empty set. However, its relative interior is

 $\operatorname{ri}(I^{2})=\{(x,y,0)\mid 0

since $\operatorname{Aff}(I^{2})$ is the $x$-$y$ plane $\{(x,y,0)\mid x,y\in\mathbb{R}\}$. Next, consider the closed unit cube

 $I^{3}:=\{(x,y,z)\mid 0\leq x,y,z\leq 1\}$

in $\mathbb{R}^{3}$. The interior and the relative interior of $I^{3}$ are the same:

 $\operatorname{int}(I^{3})=\operatorname{ri}(I^{3})=\{(x,y,z)\mid 0

Remarks.

• As another example, the relative interior of a point is the point, whereas the interior of a point is $\varnothing$.

• It is true that if $T\subseteq S$, then $\operatorname{int}(T)\subseteq\operatorname{int}(S)$. However, this is not the case for the relative interior operator $\operatorname{ri}$, as shown by the above two examples: $\varnothing\neq I^{2}\subseteq I^{3}$, but $\operatorname{ri}(I^{2})\cap\operatorname{ri}(I^{3})=\varnothing$.

• The companion concept of the relative interior of a set $S$ is the relative boundary of $S$: it is the boundary of $S$ in $\operatorname{Aff}(S)$, denoted by $\operatorname{rbd}(S)$. Equivalently, $\operatorname{rbd}(S)=\overline{S}-\operatorname{ri}(S)$, where $\overline{S}$ is the closure of $S$.

• $S$ is said to be relatively open if $S=\operatorname{ri}(S)$.

• All of the definitions above can be generalized to convex sets in a topological vector space.

Title relative interior RelativeInterior 2013-03-22 16:20:07 2013-03-22 16:20:07 CWoo (3771) CWoo (3771) 13 CWoo (3771) Definition msc 52A07 msc 52A15 msc 51N10 msc 52A20 relative boundary relatively open