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Suppose we have two sets of data, $X$ and $Y$, each of which contains $10$ positive numbers. Now let us order the data sets $$X=\left\{ x_{1},\cdots,x_{10}\right\},\quad x_{1}\ge\cdots\ge x_{10}>0$$ and $$Y=\left\{ y_{1},\cdots,y_{10}\right\},\quad y_{1}\ge\cdots\ge y_{10}>0$$ and define $d:=\sum_{k=1}^{10}\left|x_{i_{k}}-y_{j_{k}}\right|$, that is the sum of the distances of the numbers in pairs from the two data sets.

How to prove that $d$ achieves its minimum when $i_{k}=j_{k}=k$ for $1\le k\le10$, or is there any counter example if it is not true?

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    How are $i_k$ and $j_k$ defined? Do you already have some ordering for $X$ and $Y$ and then assign a new one to them?2012-01-01
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    @Alex. ${i_1,\cdots,i_{10}}$ is a permutation of ${1, 2, \cdots, 10}$, similarly for ${j_1,\cdots,j_{10}}$.2012-01-01
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    Have you tried applying the triangle inequality ($|a+b|\leq |a|+|b|$)?2012-01-01
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    @Alex. Yes, I have tried that. But I couldn't prove or disprove it.2012-01-01

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