Proving minimum of distances exists

Question

My first stackexchange post! I'm ready to be flamed haha. ;)

My problem: For some n points, prove that there exists a point p that minimizes the sum of the distances between p and each of the n points.

At first, I thought this would be easy to prove, as it's pretty intuitive. It's easy to see there's no maximum for the distance, so how do I prove that there must be some minimum?

score 0 · Answer 1 · answered Nov 12 '17 at 20:32

0

HINT

Let $D$ be a disk containing all the points $\{M_i\}$ and $f:D \rightarrow \mathbb{R_+}, f(P) = \sum_i |PM_i|$. Then $f$ is continuous and $D$ is compact, therefore the image of $f$ is compact, which means bounded.

answered Nov 12 '17 at 20:32

I know it should be intuitive, but what says P can't be outside disk D? – ekim Nov 12 '17 at 20:41
@ekim Take the disk big enough – Nov 12 '17 at 20:42
Oh is that what you meant by f:D→R+? – ekim Nov 12 '17 at 20:43
argh im still confused where to go from here... – ekim Nov 12 '17 at 22:30
Is the next step to take the fact that f is continuous and use the extreme value theorem on this compact image of f to determine that a minimum S(P) exists? – ekim Nov 12 '17 at 22:54
@ekim Yes ..... – Nov 13 '17 at 18:15

Proving minimum of distances exists

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