Prove that if $B(x,r)$ and $B(x',r')$ are disjoint $\Longleftrightarrow d(x,x') \ge r+r'$

Question

Assuming that $d(x,x') \ge r+r'$, and proving that they are disjoint is easy. It's the other side that I'm having difficulty with. This seems like a really easy problem, but i'm having difficulty proving it with basically just using the triangle inequality. Any help appreciated!

Answer 1

It is false in metric spaces in general. For example, suppose only two points exist in the space and the distance between them is $1$. Then the open balls of radius $3/4$ about the two points are disjoint even though $3/4+3/4>1$.

So you need some additional assumptions about the space involved beyond what things like the triangle inequality can give you.

If it's a Euclidean space, you can look at the set $\{wx+(1-w)x' : 0\le w\le1\}$, which is just the segment with endpoints $x$ and $x'$. If the two open balls are disjoint, you should be able to find a point $y$ on that segment whose distance from $x$ is more than $r$ and whose distance from $x'$ is more than $r'$, and you should be able to show that the distance from $x$ to $y$ plus the distance from $y$ to $x'$ is the distance from $x$ to $x'$.