Question about relative interiors and convexity

Question

Suppose that $C\subseteq \mathbb{R}^n$, such that $\operatorname{ri} C\neq \emptyset$ is convex and $\operatorname{cl} C$ is convex. Can we show that $\operatorname{cl}(\operatorname{ri}C)=\operatorname{cl}C$? If not, is there a counterexample?

Answer 1

Any open convex $C$ is a counterexample [to the original version with desired conclusion $\operatorname{ri}(\operatorname{cl} C) = \operatorname{cl} C$].

If you meant $\operatorname{ri}(\operatorname{cl} C) = \operatorname{ri} C$, then a counterexample is $[0,1]\cup(\mathbb{Q}\cap[1,2])$, whose (relative) interior is $(0,1)$ and whose closure is $[0,2]$.

[And that's also a counterexample for the current version, with $\operatorname{cl}\operatorname{ri} = \operatorname{cl}$.]