- E14
[17D] Topics:projection.Difficulty:*. Note:This is the well-known ”projection theorem”, which holds for \(A\) convex closed in a Hilbert space; if \(A⊂ ℝ^ n\) then the proof is simpler, and it’s a useful exercise..
Given \(A⊂ ℝ^ n\) closed convex non-empty and \( z∈ ℝ^ n\), show that there is only one minimum point \(x^*\) for the problem
\[ \min _{x∈ A} \| z-x\| ~ . \]Show that \(x^*\) is the minimum if and only if\[ ∀ y∈ A , ⟨ z-x^*, y-x^* ⟩≤ 0 ~ ~ . \]\(x^*\) is called ”the projection of \(z\) on \(A\)”.![\includegraphics[width=0.9\linewidth ]{UUID/1/7/F/blob_zxx}](/UUID/EDB/17D/html/images/img-0001.png)
(Note that this last condition is simply saying that the angle must be obtuse.)
Solution 1
EDB — 17D
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Authors:
"Mennucci , Andrea C. G."
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