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,

$${\displaystyle \operatorname {relint} (S):=\{x\in S:{\text{ there exists }}\epsilon >0{\text{ such that }}N_{\epsilon }(x)\cap \operatorname {aff} (S)\subseteq S\},}$$

where ${\displaystyle \operatorname {aff} (S)}$ is the affine hull of ${\displaystyle S,}$ and ${\displaystyle N_{\epsilon }(x)}$ 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.

For any nonempty convex set ${\displaystyle C\subseteq \mathbb {R} ^{n}}$ the relative interior can be defined as

$${\displaystyle \operatorname {relint} (C):=\{x\in C:{\text{ for all }}y\in C,{\text{ there exists some }}\lambda >1{\text{ such that }}\lambda x+(1-\lambda )y\in C\}.}$$

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