View source for Relative interior

{{Short description|Generalization of topological interior}}
In [[Mathematics|mathematics]], the '''relative interior''' of a [[Set (mathematics)|set]] is a refinement of the concept of the [[Interior (topology)|interior]], which is often more useful when dealing with low-dimensional sets placed in higher-dimensional spaces.

Formally, the relative interior of a set <math>S</math> (denoted <math>\operatorname{relint}(S)</math>) is defined as its [[Interior (topology)|interior]] within the [[Affine hull|affine hull]] of <math>S.</math>{{sfn|Zălinescu|2002|pp=2-3}} In other words,
<math display=block>\operatorname{relint}(S) := \{ x \in S : \text{ there exists } \epsilon > 0 \text{ such that } B_\epsilon(x) \cap \operatorname{aff}(S) \subseteq S \},</math>
where <math>\operatorname{aff}(S)</math> is the affine hull of <math>S,</math> and <math>B_\epsilon(x)</math> is a [[Ball (mathematics)|ball]] of radius <math>\epsilon</math> centered on <math>x</math>. Any metric can be used for the construction of the ball; all metrics define the same set as the relative interior.

A set is '''relatively open''' iff it is equal to its relative interior. Note that when <math>\operatorname{aff}(S)</math> is a [[Subspace topology|closed subspace]] of the full vector space (always the case when the full vector space is finite dimensional) then being '''relatively closed''' is equivalent to being closed.

For any [[Convex set|convex set]] <math>C \subseteq \mathbb{R}^n</math> the relative interior is equivalently defined as<ref>{{cite book |author=Rockafellar, R. Tyrrell |title=Convex Analysis |publisher=Princeton University Press |year=1997 |isbn=978-0-691-01586-6 |location=Princeton, NJ |page=47 |orig-year=First published 1970}}</ref><ref>{{cite book |author=Dimitri Bertsekas |title=Nonlinear Programming |publisher=Athena Scientific |year=1999 |isbn=978-1-886529-14-4 |edition=2nd |location=Belmont, Massachusetts |page=697 |author-link=Dimitri Bertsekas}}</ref>
<math display="block">\begin{align}\operatorname{relint}(C) &:= \{x \in C : \text{ for all } y \in C, \text{ there exists some } \lambda > 1 \text{ such that } \lambda x + (1 - \lambda) y \in C\}\\
&= \{x \in C : \text{ for all } y\neq x \in C, \text{ there exists some } z \in C \text{ such that } x\in (y,z)\}.
\end{align}</math>
where  <math> x\in (y,z) </math> means that there exists some <math> 0< \lambda < 1 </math> such that <math> x=\lambda z + (1 - \lambda) y </math>.
== Comparison to interior ==

* The interior of a point in an at least one-dimensional ambient space is empty, but its relative interior is the point itself.

* The interior of a line segment in an at least two-dimensional ambient space is empty, but its relative interior is the line segment without its endpoints.

* The interior of a disc in an at least three-dimensional ambient space is empty, but its relative interior is the same disc without its circular edge.

== Properties ==

{{Math theorem
| math_statement = If <math>A\subset \R^n</math> is nonempty and convex, then its relative interior <math>\mathrm{relint}(A)</math> is the union of a nested sequence of nonempty compact convex subsets <math>K_1\subset K_2\subset K_3\subset\cdots \subset \mathrm{relint}(A)</math>.
}}

{{Math proof|title=Proof|proof=  

Since we can always go down to the affine span of <math>A</math>, WLOG, the relative interior has dimension <math>n</math>. Now let <math> K_j \equiv [-j,j]^n \cap \left\{ x \in \text{int}(K) : \mathrm{dist}(x, (\text{int}(K))^c) \ge \frac{1}{j} \right\} </math>.
}}

{{Math theorem
| math_statement = Here "+" denotes Minkowski sum. 

* <math>\mathrm{relint}(S_1) + \mathrm{relint}(S_2) \subset \mathrm{relint}(S_1 + S_2)</math> for general sets. They are equal if both <math>S_1, S_2</math> are also convex.

* If <math>S_1, S_2</math> are convex and relatively open sets, then <math>S_1 + S_2</math> is convex and relatively open.
}}

{{Math theorem
| math_statement = Here <math>\mathrm{Cone}</math> denotes positive cone. That is, <math>\mathrm{Cone}(S) = \{rx: x\in S, r > 0\}</math>.

* <math> \mathrm{Cone}(\mathrm{relint}(S)) \subset \mathrm{relint}(\mathrm{Cone}(S))</math>. They are equal if <math>S</math> is convex.
}}

== See also ==

* {{annotated link|Interior (topology)}}
* {{annotated link|Algebraic interior}}
* {{annotated link|Quasi-relative interior}}

== References ==

{{reflist}}
{{reflist|group=note}}
{{reflist|group=proof}}

* {{Zălinescu Convex Analysis in General Vector Spaces 2002}} <!-- {{sfn|Zălinescu|2002|pp=}} -->

==Further reading==

* {{cite book|title=Convex Optimization
|last1=Boyd
|first1=Stephen
|author2=Lieven Vandenberghe
|year=2004
|publisher={{wipe|Cambridge University Press}}
|location=Cambridge
|url=https://web.stanford.edu/~boyd/cvxbook/
|page=23
|isbn=0-521-83378-7
}}

<!--
fr:Intérieur relatif d'un convexe
-->

{{Functional analysis}}

[[Category:Topology]]

{{Sourceattribution|Relative interior}}