I am trying to prove that aff $ C \subset $ aff ri $C$ for a convex set $C$ (i.e., affine hull of the set $C$ is a subset of the affine hull of its relative interior). My question is, does $x\in C$ but $x\notin $ ri $C$ imply that $x$ is a boundary point of $C$? Also, does it matter if $C$ is convex or not?
Relative interior and limit points
convex-analysis
Best Answer
Yes. If $x \in C$, but $x \notin \operatorname{ri} C$, then $x$ is in the boundary of $C$. Prove the contrapositive: if $x \in C$ is not in the boundary of $C$, then $x \in \operatorname{int} C$. This means $C$ contains a ball, so the affine hull of $C$ is the full space. In this case, $\operatorname{int} C = \operatorname{ri} C$. Hence, $x \in \operatorname{ri} C$.
The point of the relative interior is that boundary and interior aren't descriptive enough for general convex sets, or indeed, other (non-empty) sets. As soon as the affine hull of a non-empty set $C$ is not the full space, then $\operatorname{int} C = \emptyset$ and the boundary of $C$ covers $C$.
(Also, you should specify the space you're working in. In a general normed linear space, the result you're trying to prove is false.)