Condition for connectedness of two open ball

Question

What is the condition on $p, q, r_1, r_2 $ such that $B(p,r_1) \cup B(q, r_2)$ is connected in $\mathbb{R^n}$?

We know that if the intersection is non empty then union is connected but how to find condition? Plese help.

Answer 1

The intersection point lays in $B(p,r_1)$ and $B(q,r_2)$, hence it has distance of $p$ less than $r_1$ and of $q$ less than $r_2$. Using triangle inequality, what does this mean to the distance of $p$ and $q$.

To show this is enough look at the line joining $p$ and $q$ and choose a point in appropriate distance of $p$. On this line the triangle inequality is an equality, hence if you chose the appropriate distance, it should still lay in $B(q,r_2)$ (if the distance of $p$ and $q$ is small enough). (What is small enough? :-) )