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?
2026-03-27 10:15:54.1774606554
Relative interior of convex sets
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