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I have a data $X(i)=i$, where $i \in ${1,2,3,4,5,6}, i want to find $u$ that minimises $max|X(i)-u|$. How to solve this problem, can you give a proof?

anfjnlnl
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    Hint: take $u=\frac{\max_i X(i)+\min_i X_i}{2}$, i.e. right in the middle between the highest and lowest of $X(i)$'s. (Why?) If $X(i)=i$, then $u=3.5$. –  Apr 11 '21 at 10:36
  • @StinkingBishop can you please prove this result or post the link which has proof – anfjnlnl Apr 11 '21 at 10:46
  • I would prefer if you can show that you have made some effort yourself. This is not a "solve a problem for me" site. Cf. "How do I ask a good question": https://math.stackexchange.com/help/how-to-ask . –  Apr 11 '21 at 16:32
  • Having said that, there is another hint (or, rather, two): (a) Let $m=\min_i X(i), M=\max_i X(i)$. Can you prove that, for any $u\in\mathbb R$ you have $|m-u|\ge \frac{M-m}{2}$ or $|M-u|\ge \frac{M-m}{2}$ ? (Hint: use the triangle inequality.) (b) Can you prove that $\max_i|X(i)-u|=\frac{M-m}{2}$ if $u$ is the number $\frac{M+m}{2}$? –  Apr 11 '21 at 16:36
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    (Also, I don't believe your problem is about $\max(X(i)-u)$ but about $\max|X(i)-u|$. Can you please check? With such an elementary omission, I am not sure I am really helping you with those hints above.) –  Apr 11 '21 at 16:39
  • @StinkingBishop thanks man – anfjnlnl Apr 12 '21 at 17:46
  • https://math.stackexchange.com/questions/1179866/solve-textminimize-max-x-a-i-i-1-n – RobPratt Apr 13 '21 at 02:14

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