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I want to know if for a given nonempty closed convex subset $X$ of a finite dimensional normed space it holds $\operatorname{ri} X \subset \operatorname{int} Y$ ($\operatorname{ri}$ denotes the relative interior and $\operatorname{int}$ the interior), then can one infer $X \subset Y$?

karlabos
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No: counterexamples in $\Bbb R$ are $X=[0,1]$ and $Y=(0,2)$. In fact, $$\operatorname{relint} X=\operatorname{int} X=(0,1)\subsetneq \operatorname{int} Y=Y$$ but $X\nsubseteq Y$.

Added: It is, however, true if $Y$ is convex and closed, since for a convex subset $Y$ of a finite dimensional vector space it holds $\overline Y=\overline{\operatorname{relint} Y}$ (of course, if $\operatorname{int} Y\ne\emptyset$, then $\operatorname{int}Y=\operatorname{relint}Y$)

  • I see, thanks. What if I add that $Y$ is also closed and convex. Does it hold then or is there other counterexample? – karlabos Sep 10 '16 at 22:54