Sum of balls of convex set.

Question

Let $V$ be an Euclidean space, with a non-empty convex set $Z$. Let $r>0$. Prove that the set $\cup_{x \in Z} \ \ B(x,r) \supset Z$ is also convex in $V$.

There are plenty of proofs when I have a sum of finite (or $\mathbb{N} $ elements) but I can't find any hints on how to tackle this problem. Any hints are greatly appreciated.

Answer 1

Pick two points $u,v$ in the set $K = \cup_{x\in Z} B(x,r)$ and $x,y\in Z$ such that $$ \|u-x\|