# Finding a unit vector $\xi$ such that $H_{K_1}(-\xi) + H_{K_2}(\xi) < 0$ for disjoint compact convex sets $K_1, K_2$

The supporting function of a compact convex set $K$ is defined as $$H_K(x) := \sup_{t\in K} \langle t,x \rangle,$$ where $\langle \cdot, \cdot \rangle$ is the usual inner product.

I'm trying to understand the proof of a certain theorem, in particular it is claimed that if $K_1$ and $K_2$ are two disjoint compact convex sets, then there exists a real unit vector $\xi$ such that $$H_{K_1}(-\xi) + H_{K_2}(\xi) = \sup_{t\in K_2} \langle t,\xi \rangle - \inf_{t\in K_1} \langle t,\xi \rangle < 0.$$ I just cannot wrap my head around it. What is the easiest way to see this?

What you want is $\xi$ nonzero such that $$\sup\langle K_2-K_1,\xi\rangle < 0$$ But this is true because $K_2-K_1$ is compact and convex, and $0$ is not in $K_2-K_1$, i.e., $K_1\cap K_2=\varnothing$.