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How to prove the affirmation?:

If $K_1$ and $K_2$ are nonempty, nonintersecting, convex and open sets, there exists a closed hyperplane $M$ such that $K_1$ and $K_2$ are strictly on opposite sides of M.

Exists a version of Hahn-Banach Theorem that said: Let $A ⊂ E$ and $B ⊂ E$ be two nonempty convex subsets such that $A ∩B = ∅$. Assume that one of them is open. Then there exists a closed hyperplane that separates $A$ and $B$. (But not necessary in the sense strictly)

Vivi
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  • If $A$ is open and convex, and if it lies in a closed half-space, then it even lies in the open half-space. If you apply this to $A$ and $B$, your desired result follows. – gerw Nov 27 '15 at 09:33

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If $F\subset G$, then the interior of $F$ (denoted $\operatorname{int}F$) is contained in $\operatorname{int}G$. As a special case: if $F\subset G$ and $F$ is open, then $F\subset\operatorname{int}G$. With this in mind, the following comment by gerw completes the solution.

If $A$ is open and convex, and if it lies in a closed half-space, then it even lies in the open half-space. If you apply this to $A$ and $B$, your desired result follows.

Community
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