Prove convexity of set I've had a hard time proving this statement. The objective is to prove that the set $M$ is convex where $f(y)$ can be any function. The task is to prove it using triangle inequality. I've looked at threads like Proving Convexity of an Open Disk but i still can't wrap my head around it. If anyone could give me a hint in the right direction i would be very grateful.
$$M = \{\, x\,\big| \, ||x-y|| \leq f(y) \, \text{ for all } y \in S\,\} \quad \text{where } S \subseteq \mathbb{R}^n$$
Best Regards

Another hint:  $M$ is the intersection of a certain collection of closed balls.  
