I have a basic question about the independence of two events.
Let $x$ be some fixed point in the interior of some set (say it's a convex bounded set $C$ in $\mathbb{R}^2$ or something). Choose $a, b, y$ uniformly at random and independently from the interior of $C$. Are the events that $||x - a|| < ||x - b||$ and $||y - a|| < ||y-b||$ independent? That is, is it true that $$\Pr(||x-a|| < ||x-b||\text{ and } ||y-a|| < ||y-b||) \\ = \Pr(||x-a|| < ||x-b||)\cdot\Pr(||y-a|| < ||y-b||) = 1/2?$$ I would really think so, but I'm not sure how to rigorously show this. Maybe it's enough to say that $||x-a||, ||y-a||$ are independent random variables?
What if now $a, b$ are chosen uniformly at random from some subset $C'\subset C$ (again, some "nice" subset), and $y$ is as before chosen uniformly at random from $C$. Are the events independent?