I have the following problem about compact convex sets.
Let $K\subset\mathbb{R}^n$ be a compact convex set with nonempty interior. Assume that $A_1,\dots,A_m$ are open half spaces that cover $\partial K$ and set $P:=\mathbb{R}^n\setminus \bigcup_{i=1}^m A_i$. Show that $P$ is a subset of $K$ and hence $P$ is a polytope.
I was trying to show that if $x\notin K$ then there exists some $A_i$ so that $x\in A_i$. Even though this fact is intuitively true, but I am not sure how to make it rigorous. I guess we need something like $\partial K$ ''enclosing'' $K$ and the condition that $K$ has interior point somewhere.
Can someone help me? Thanks for any hint!
I think this assertion is wrong. Consider any compact convex set $K$ and then take one halfspace $A_1$ which contains all of $K$ and all of $\partial K$ (which exists since $K$ is compact). Now $P$ is a closed halfspace and non-empty and $K \cap P = \emptyset$, so $P$ is not a subset of $K$.