I have a problem that I'm not sure how to solve.
Here's the problem. Suppose I have a one-dimensional space with people distributed some way over that space. They could be normally distributed, they could be uniformly distributed, they could be distributed any which way. Now suppose I want to place two points on the space such that I minimize the average distance between each person and the point closest to them. So, I want to place the two points so people are, on average, as close as possible to their closest point. Where should I put the points?
I was thinking that the solution is straightforward. First, locate the person in the median position. This is the guy in the middle of the distribution. Now split the people into those left of the median and those right of the median. You should put one point at the median position on the the left and one at the median on the right. But (a) I'm not certain this is the right solution for every distribution of people, and (b) I'm not sure how to prove this is the solution. Any help would be greatly appreciated!
