# 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 S (denoted ${\displaystyle \operatorname {relint} (S)}$) is defined as its interior within the affine hull of S.[1] In other words,

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

where ${\displaystyle \operatorname {aff} (S)}$ is the affine hull of 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 sets ${\displaystyle C\subseteq \mathbb {R} ^{n}}$ the relative interior can be defined as

${\displaystyle \operatorname {relint} (C):=\{x\in C:\forall {y\in C}\;\exists {\lambda >1}:\lambda x+(1-\lambda )y\in C\}.}$[2][3]