Let $X$ be a Hausdorff real locally convex topological vector space and $K$ be a nonempty compact subset. Bauer's maximum principle states that a convex upper semicontinuous function $f\colon K \to \mathbb R$ attains its maximum on $K$ at an extreme point of $K$.
My question is the converse: given an extreme point $x$ of $K$, does there always exist a convex upper semicontinuous function $f\colon K\to\mathbb R$ such that $f(x) > f(y)$ for all $y\in K \setminus \{x\}$?
I have found that if $K$ has nonempty interior, the Hahn-Banach convex separation theorem concludes there is indeed such a function (which is even a continuous linear form on $X$). At the same time, this is a restrictive hypothesis.