Let $K \subset \mathbb{R}^d$ be compact and convex. Consider a point $x \in K$ such that $\lVert x \rVert$ is maximized and let $H$ be the hyperplane with equation $\langle y, x \rangle = \lVert x \rVert$. Is it true that $H \cap K = \{ x \}$ i.e. does $H$ intersect $K$ only at $x$? I'm unsure that $\forall a \in K$, $\langle a, x \rangle < \langle x, x \rangle$.
What I'm trying to show is that every compact, convex set has a $0$-dimensional face.
This is false. For instance if $0\in K$ then there are lots of examples where $H\cap K$ contains way more points than only $x$.
For your more general question you might like to have a look at Krein-Milman's theorem.