This exercise comes from Bazaraa Linear Programming and Network Flows book :
Consider the problem of locating a new machine to an existing layout
consisting of four machines. These machines are located at the following coordinates in two-dimensional space: $(3,1) ,(0,-3) ,(-2,2)$ and $(1,4)$. Let the coordinates of the new machine be $(x_1,x_2)$ .
The sum of the distances from the new machine to the four machines is minimized. Use the street distance (also known as Manhattan distance or rectilinear distance); for example, the distance from $(x_1,x_2)$ to the first machine located at (3,1) is $|x_1 - 3| + |x_2 - 1|$·
the solution is definitely minimized $$|x_1 - 3| + |x_2 - 1| + |x_1 - 0| + |x_2 + 3| + |x_1 + 2| + |x_2 - 2| + |x_1 - 1| + |x_2 - 4|$$
But a solution guide I saw adds this :
$$|x_1 - 3| + |x_1 - 0| + |x_1 + 2| + |x_1 - 1| \geq 6$$ $$|x_2 - 1| + |x_2 + 3| + |x_2 - 2| + |x_2 - 4| \geq 10$$
I know those are sum of absolute values of specified points' location. But I can't figure out the whole reason of above solution. Has anyone figured out?