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 relint ⁡ ( S ) ) is defined as its interior within the affine hull of S. In other words,

where aff ⁡ ( S ) is the affine hull of S, and N ϵ ( x ) is a ball of radius ϵ 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 ⊆ R n the relative interior can be defined as