As defined in Convex Optimization written by Stephen Boyd & Lieven Vandenberghe, both interior and relative interior seems to describe a same thing: a set that peels away its boundary points. So, what on earth is the difference between these two concepts ?
Here are the definitions of these two concepts from Convex Optimization:
interior: The set of all points interior to $C$ is called the interior of $C$
relative interior: its interior relative to the affine hull of $C$
Best Answer
The relative interior of a set is the interior of the set when it is viewed as a subset of the affine space it spans.
For example, the interior of the segment connecting $(0,0)$ to $(1,1)$ in the plane is empty, but the relative interior is the open segment with those endpoints.