I have to solve
$$\text{Minimize} \max\{|x-a_i|, i=1,..,n\}$$
For $a_1 \leq a_2 \leq ...\leq a_n$
My intuition says that this x is a point in the middle of the $a_i's$ but I am not sure that it is correct and how can I prove this.
Thanks!
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over what do you take the min? – Surb Mar 07 '15 at 18:58
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I edited. The question is exactly how it is in the book – Giiovanna Mar 07 '15 at 19:00
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... it should be written somewhere where does $x$ live. – Surb Mar 07 '15 at 19:00
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Well, I think it says to find the x that minimize the maximum distance between x and the given numbers – Giiovanna Mar 07 '15 at 19:01
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The $\max$ will always take its value either for $i=1$ or $i = n$. So the minimize will make these two scenarios equal (as small as possible), so the minimum is indeed at $(a_n + a_1)/2$, i.e. in the "middle" in some sense like you suspected.
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