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How many integral values of $(X,Y)$ are there such $$((|X-L_1|+|Y-M_1|)+(|X-L_2|+|Y-M_2|)+\cdots+(|X-L_n|+|Y-M_n|))$$ is minimized?

$10^{-6} < X < 10^6$

$10^{-6} < Y < 10^6$

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1 Answer 1

up vote 2 down vote accepted

You want X to be a median of the L's and Y to be a median of the M's. For an even count, this can be any value between the central two. This is stated, but not proved, in Wikipedia's article on the median under an optimality property.

To see why the median is the right answer, if X is greater than all the L's and decreases by 1, the total will decrease by the number of L's. The total will still decrease with decreasing X until there are as many L's above as below X. That is the definition of median.

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When $N$ is odd, there is a unique median, so a unique $X$ and a unique $Y$. –  Gerry Myerson May 3 '12 at 5:56

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