Relative interior of convex sets It is known that whenever a set $C$ in $\mathbb{R}^n$ is convex then $\text{cl}(\text{ri}\hspace{0.1cm} C)=\text{cl}\hspace{0.1cm} C$ and $\text{ri}(\text{cl}\hspace{0.1cm}C)=\text{ri}\hspace{0.1cm}C$ where $\text{cl}\hspace{0.1cm}C$ and $\text{ri}\hspace{0.1cm}C$ are the closure and the relative interior of a set $C$. Why wouldn't this statement hold in general? What are counterexamples when $C$ is not convex?
Consider a sphere $C$ (without interior) in $\mathbb{R}^2$. How would you sensibly define "relative interior"? The set $C$ should serve as a counterexample.
Consider a sphere $C$ (without interior) in $\mathbb{R}^2$. How would you sensibly define "relative interior"? The set $C$ should serve as a counterexample.