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 ) is defined as its interior within the affine hull of S. In other words, where is the affine hull of S, and 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. A set is relatively open iff it is equal to its relative interior. Note that when is a closed subspace of the full vector space (always the case when the full vector space is finite dimensional) then being relatively closed is equivalent to being closed. For any convex set the relative interior is equivalently defined as where x\in (y,z) means that there exists some such that.

Comparison to interior

Properties