hyperplane separating point in convex hull from compact set

Question

If a point is inside the convex hull of a compact set, then could it be proved that every hyperplane with the compact set on 1 side will always have that point $x$ on the same side?

Answer 1

Suppose $A$ is a set such that $\phi(a) \le \alpha $ for all $a \in A$ where $\phi$ is a linear functional and $\alpha$ is some constant.

Note that the set $H = \{x | \phi(x) \le \alpha \}$ is convex and $A \subset H$. Since $\operatorname{co} A$ is the intersection of all convex sets containing $A$ we see that $\operatorname{co} A \subset H$ and hence any point in $\operatorname{co} A$ lies in the 'same side' of the hyperplane $\{x | \phi(x) = \alpha \}$ as $A$.