If $\emptyset \neq X$ is closed and $\operatorname{ri} X \subset \operatorname{int} Y$ then $X \subset Y$?

Question

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 $X \subset Y$?

Answer 1

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 $\operatorname{int}Y=\operatorname{relint}Y$)