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 $$I^2:=\lbrace (x,y,0)\mid 0\le x, y\le 1\rbrace$$ in $\mathbb{R}^3$ . Its interior is $\varnothing$ , the empty set. However, its relative interior is $$\operatorname{ri}(I^2)=\lbrace (x,y,0)\mid 0< x,y< 1\rbrace,$$ since $\operatorname{Aff}(I^2)$ is the $x$ -$y$ plane $\lbrace (x,y,0)\mid x,y\in\mathbb{R}\rbrace$ . Next, consider the closed unit cube $$I^3:=\lbrace (x,y,z)\mid 0\le x, y, z\le 1\rbrace$$ in $\mathbb{R}^3$ . The interior and the relative interior of $I^3$ are the same: $$\operatorname{int}(I^3)=\operatorname{ri}(I^3)=\lbrace (x,y,z)\mid 0< x,y,z< 1\rbrace.$$

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.