You can prove it by induction.. Very standard high school stuff.

First we argue that we can look at 1-d. Because the two dimension can actually be computed seperately.

n =1, it's the median point. We know this is the best point.

n=2, there are two middle point x0,x1, (x0<=x1) pick any one. We also know this is the best point. Actually the min distance is |x0-x1|. You can pick any point between x0 to x1. pick another point x<x0 or x>x1, you will pay extra.

n=1,2 are our base cases, we prove the median point is the best point.

For n>=3. We pick the leftmost point x0 and rightmost point xn. so we reduce to n-2, in the remaining n-2 points we already prove the median point of the n-2 points is the best.. And this median point is also the median point of the n points..

I guess you got my point.