EDB — 17J

[17J] Topics:separation. Difficulty:*.

This result applies in very general contexts, and is a consequence of Hahn–Banach theorem (which makes use of Zorn’s Lemma); if \(A⊂ ℝ^ n\) it can be proven in an elementary way, I invite you to try.

Given \(A⊂ ℝ^ n\) open convex non-empty and \( z∉ A\), show that there is a hyperplane \(P\) separating \( z\) from \(A\), that is, \(z\in P\) while \(A\) is entirely contained in one of the two closed half-spaces bounded by the hyperplane \(P\). Equivalently, in analytical form, there exist \(a∈ℝ,v∈ℝ^ n,v≠ 0\) such that \(⟨ z,v⟩=a\) but \(∀ x∈ A,⟨ x,v⟩{\lt}a\); and

The hyperplane \(P\) thus defined is called supporting hyperplane of \(z\) for \(A\).

There are (at least) two possible proofs. A possible proof is made by induction on \(n\); we can assume without loss of generality that \( z=e_ 1=(1,0\dots 0),0∈ A, a=1\); keep in mind that the intersection of a convex open sets with \(ℝ^{n-1}× \{ 0\} ⊂ℝ^ n\) is an open convex set in \(ℝ^{n-1}\); this proof is complex but does not use any prerequisite. A second proof uses [176]↺↻ and [17H]↺↻ if \(z∉∂ A\); if \(z∈∂ A\) it also uses [178]↺↻ to find \((z_ n)⊂ {{(A^ c)}^\circ }\) with \(z_ n→ z\) .