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. 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 relint ⁡ ( S ) {displaystyle operatorname {relint} (S)} ) is defined as its interior within the affine hull of S. In other words, where aff ⁡ ( S ) {displaystyle operatorname {aff} (S)} is the affine hull of S, and N ϵ ( x ) {displaystyle N_{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. For any nonempty convex set C ⊆ R n {displaystyle Csubseteq mathbb {R} ^{n}} the relative interior can be defined as