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

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

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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 $\mathrm{ri}({I}^{2})\cap \mathrm{ri}({I}^{3})=\mathrm{\varnothing}$.
 •

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

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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  20130322 16:20:07 
Last modified on  20130322 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 