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.

Title	relative interior
Canonical name	RelativeInterior
Date of creation	2013-03-22 16:20:07
Last modified on	2013-03-22 16:20:07
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	13
Author	CWoo (3771)
Entry type	Definition
Classification	msc 52A07
Classification	msc 52A15
Classification	msc 51N10
Classification	msc 52A20
Defines	relative boundary
Defines	relatively open