Consider placing countably infinitely many points labeled $S_i$ randomly over $\mathbb{R}^2$, with asymptotic density points/area $µ$.
Then, what is the largest $r$ such that we can find a a continuous curve with no finite square containing it, and such that all points $S_i$ have distance atleast $r$ to it?
And the same question for $\mathbb{R}^3$?