EDB β€” 17D

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E14

[17D] Topics:projection.Difficulty:*. Note:This is the well-known ”projection theorem”, which holds for A convex closed in a Hilbert space; if AβŠ‚β„n then the proof is simpler, and it’s a useful exercise..

Given AβŠ‚β„n closed convex non-empty and zβˆˆβ„n, show that there is only one minimum point xβˆ— for the problem

minx∈Aβ€–zβˆ’xβ€– .
Show that xβˆ— is the minimum if and only if
βˆ€y∈A,⟨zβˆ’xβˆ—,yβˆ’xβˆ—βŸ©β‰€0  .
xβˆ— is called ”the projection of z on A”.
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(Note that this last condition is simply saying that the angle must be obtuse.)

Solution 1

[17G]

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