, 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 ) is defined as its interior within the affine hull of S. In other words,
is the affine hull of S
is a ball
. Any metric can be used for the construction of the ball; all metrics define the same set as the relative interior.