relative interior

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

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

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

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

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

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

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

Remarks.

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

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

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

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

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