# 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. Intuitively, the relative interior of a set contains all points which are not on the "edge" of the set, relative to the smallest subspace in which this set lies.

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

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

where $\operatorname{aff}(S)$ is the affine hull of S, and $N_\epsilon(x)$ is a ball of radius $\epsilon$ centered on $x$. 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 $C \subseteq \mathbb{R}^n$ the relative interior can be defined as

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