Of course, a compact set has Supporting Hyperplanes (SH), a convex set with boundary has SH. Those conditions are sufficient, but not necessary.
What is the sufficient and necessary condition for a given set to have at least one supporting hyperlane?
Does the set $\{(x,y)|y>tan(x)\}$ has a supporting hyperplane?
For example, the set $\{(x,y)|x,y\in\mathbb Z\}$ does not have a supporting hyperplane, because any line in $\mathbb R^n$ cannot bound the set.
$\{(x,y)|x,y\in\mathbb N^*\}$ does have one where $\mathbb N^*$ is the set of positive naturals.