Interior and convex compact sets

Question

Let $A$, $B$ and $C$ be convex, compact, nonempty sets defined on a metric space, with $B \subseteq A \cap C$. I have the following implications:

i) $x \in A \wedge x \in C \implies x \in B$

ii) $x \in \textrm{int}(B) \implies x \in A$

where $\textrm{int}$ denotes the topological interior. Not able neither to prove nor to disprove that $B \subseteq A$, maybe it is trivial...

Answer 1

This is false even in $\mathbb R^{2}$. Let $A,B,C$ be the line segments for the origin to $(1,0), (2,0)$ and $(3,0)$ respectively. Then ii) holds vacuously since $B$ has no interior. Of course i) holds and $B$ is not a subset of $A$.