Relative interior

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 ${\displaystyle S}$ (denoted ${\displaystyle \operatorname {relint} (S)}$) is defined as its interior within the affine hull of ${\displaystyle S.}$^[1] In other words, $${\displaystyle \operatorname {relint} (S):=\{x\in S:{\text{ there exists }}\epsilon >0{\text{ such that }}B_{\epsilon }(x)\cap \operatorname {aff} (S)\subseteq S\},}$$ where ${\displaystyle \operatorname {aff} (S)}$ is the affine hull of ${\displaystyle S,}$ and ${\displaystyle B_{\epsilon }(x)}$ is a ball of radius ${\displaystyle \epsilon }$ centered on ${\displaystyle x}$. 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 ${\displaystyle \operatorname {aff} (S)}$ 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 ${\displaystyle C\subseteq \mathbb {R} ^{n}}$ the relative interior is equivalently defined as^[2][3] $${\displaystyle {\begin{aligned}\operatorname {relint} (C)&:=\{x\in C:{\text{ for all }}y\in C,{\text{ there exists some }}\lambda >1{\text{ such that }}\lambda x+(1-\lambda )y\in C\}\\&=\{x\in C:{\text{ for all }}y\neq x\in C,{\text{ there exists some }}z\in C{\text{ such that }}x\in (y,z)\}.\end{aligned}}}$$ where ${\displaystyle x\in (y,z)}$ means that there exists some ${\displaystyle 0<\lambda <1}$ such that ${\displaystyle x=\lambda z+(1-\lambda )y}$.