In mathematics, the relative interior of a set is a refinement of the concept of the interior, which is often more useful when dealing with low-dimensional sets placed in higher-dimensional spaces. Formally, the relative interior of a set S {\displaystyle S} (denoted relint ( S ) {\displaystyle \operatorname {relint} (S)} ) is defined as its interior within the affine hull of S . {\displaystyle S.} In other words,
relint ( S ) := { x ∈ S : there exists ϵ > 0 such that B ϵ ( x ) ∩ aff ( S ) ⊆ S } , {\displaystyle \operatorname {relint} (S):=\{x\in S:{\text{ there exists }}\epsilon >0{\text{ such that }}B_{\epsilon }(x)\cap \operatorname {aff} (S)\subseteq S\},}
where aff ( S ) {\displaystyle \operatorname {aff} (S)} is the affine hull of S , {\displaystyle S,} and B ϵ ( x ) {\displaystyle B_{\epsilon }(x)} is a ball of radius ϵ {\displaystyle \epsilon } centered on x {\displaystyle x} . 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 aff ( S ) {\displaystyle \operatorname {aff} (S)} is a 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 C ⊆ R n {\displaystyle C\subseteq \mathbb {R} ^{n}} the relative interior is equivalently defined as
relint ( C ) := { x ∈ C : for all y ∈ C , there exists some λ > 1 such that λ x + ( 1 − λ ) y ∈ C } = { x ∈ C : for all y ≠ x ∈ C , there exists some z ∈ C such that x ∈ ( y , z ) } . {\displaystyle {\begin{aligned}\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{aligned}}}
where x ∈ ( y , z ) {\displaystyle x\in (y,z)} means that there exists some 0 < λ < 1 {\displaystyle 0<\lambda <1} such that x = λ z + ( 1 − λ ) y {\displaystyle x=\lambda z+(1-\lambda )y} .
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
See also Interior (topology) – Largest open subset of some given set Algebraic interior – Generalization of topological interior Quasi-relative interior – Generalization of algebraic interior
References
Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.
Further reading Boyd, Stephen; Lieven Vandenberghe (2004). Convex Optimization. Cambridge: Cambridge University Press. p. 23. ISBN 0-521-83378-7.
