- E17
[17M] Prerequisites:[17J],[11T].If \(A,B\) are disjoint convex, with \(A\) open, show that there is a hyperplane separating \(A\) and \(B\), that is, there exist \(v∈ℝ^ n,v≠ 0\) and \(c∈ℝ\) such that
\begin{equation} ∀ x∈ A,⟨ x,v⟩< c \text{~ but ~ } ∀ y∈ B,⟨ y,v⟩≥ c ~ ;\label{eq:separa_ due_ convessi_ aperti} \end{equation}18moreover show that if also \(B\) is open, then you can have strict separation (i.e. strict inequality in the last term in ??).
(Hint: given \(A,B⊆ ℝ^ n\) convex nonempty, show that
\[ A-B{\stackrel{.}{=}}\{ x-y,x∈ A, y∈ B\} \]is convex; show that if \(A\) is open then \(A-B\) is open, as in [11T].)Solution 1
EDB — 17M
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Authors:
"Mennucci , Andrea C. G."
.
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