PlanetMath (more info)
 Math for the people, by the people. Sponsor PlanetMath
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Very high Entry average rating: No information on entry rating
relative interior (Definition)

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.




"relative interior" is owned by CWoo.
(view preamble | get metadata)

View style:

Also defines:  relative boundary, relatively open
Log in to rate this entry.
(view current ratings)

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 4145 times total.

Classification:
AMS MSC52A20 (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)

Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)