PlanetMath: relative interior

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relative interior

(Definition)

Let be a subset of the -dimensional Euclidean space $\mathbb{R}^n$ . The relative interior of is the interior of 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 can be illustrated in the following two examples. Consider the closed unit square

$\displaystyle I^2:=\lbrace (x,y,0)\mid 0\le x, y\le 1\rbrace$

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

$\displaystyle \operatorname{ri}(I^2)=\lbrace (x,y,0)\mid 0< x,y< 1\rbrace,$

since $\operatorname{Aff}(I^2)$ is the

plane $\lbrace (x,y,0)\mid x,y\in\mathbb{R}\rbrace$ . Next, consider the closed unit cube

$\displaystyle I^3:=\lbrace (x,y,z)\mid 0\le x, y, z\le 1\rbrace$

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

are the same:

$\displaystyle \operatorname{int}(I^3)=\operatorname{ri}(I^3)=\lbrace (x,y,z)\mid 0< x,y,z< 1\rbrace.$

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 is the relative boundary of : it is the boundary of 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 .
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.

"relative interior" is owned by CWoo.

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Also defines:

relative boundary, relatively open

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Cross-references: topological vector space, convex sets, definitions, closure, boundary, operator, point, cube, plane, empty set, square, unit, closed, difference, affine hull, interior, Euclidean space, subset
There are 3 references to this entry.

This is version 10 of relative interior, born on 2006-10-21, modified 2007-05-02.
Object id is 8466, canonical name is RelativeInterior.
Accessed 2984 times total.

Classification:

AMS MSC:	52A20 (Convex and discrete geometry :: General convexity :: Convex sets in $n$ dimensions )
	51N10 (Geometry :: Analytic and descriptive geometry :: Affine analytic geometry)
	52A15 (Convex and discrete geometry :: General convexity :: Convex sets in $3$ dimensions )
	52A07 (Convex and discrete geometry :: General convexity :: Convex sets in topological vector spaces)

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