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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
$\operatorname{ri}(I^{2})=\{(x,y,0)\mid 0<x,y<1\},$ 
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:
$\operatorname{int}(I^{3})=\operatorname{ri}(I^{3})=\{(x,y,z)\mid 0<x,y,z<1\}.$ 
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.
Mathematics Subject Classification
52A07 no label found52A15 no label found51N10 no label found52A20 no label found Forums
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