Strongly separating hyperplane

Question

Let $A$ and $B$ be compact convex sets. I want to show that there is a hyper-plane $H$ strongly separating them. I thought about the following: I'll take $a \in A$ and $b \in B$ s.t. $||a-b||$ is minimized. now we look on the segment line $[a,b]$ and take a point $c$ that lies on the segment. Now $H$ should intersect $[a,b]$ in $c$. How can I prove that such $H$ exists?

Answer 1

By defining a hyperplane $H$ that does not intersect $A$ nor $B$. Your intuition on how to define $H$ in terms of $a$ and $b$ should give you the correct hyperplane (thinking about $A$ and $B$ in $\mathbb{R}^2$ helps)

Now you just need to assume that $H$ intersects, say, $A$ and conclude that either the pair $(a,b)$ does not minimize $||a-b||$ or that $A$ is not convex.

Note that you will not need the hypothesis that $A$ and $B$ are compact once $H$ is defined, that means you probably already used it.