I want a separation result as follows; finite dimensions might not matter but is the case of present interest.
Suppose $C\subset \mathbb R^n$ is convex (not necessarily open or closed) and $0\in \partial C$ is in the boundary. Then there is a linear functional $\ell:\mathbb R^n \to \mathbb R$ that is non-negative on $C$.
My question is do we need to assume the interior $C^o$ is non-empty? The above result follows in a straightforward way with this assumption from standard classical results: $C^o$ is then a non-empty open convex set, disjoint from the convex set $\{0\}$. Then by a standard Hahn-Banach separation, there is $\ell$ linear functional and real number $\alpha$ such that $$\ell(0)=0\leq \alpha <\ell(x)$$ for all $x \in C^o$. Then by continuity, $\ell\geq 0$ on $\overline{C^o}$, which contains $C$, again if $C^o \neq \emptyset.$