I am given a set $S$ of points and I want to prove that for any point $p$ in $S$, the farthest-point voronoi region of $p$ is not empty if and only if $p$ is on the convex hull of $S$.
I denote farthest-point voronoi region of $p$ by $FVR(p)$ for simplicity.
The proof must have two parts. one part is "If $FVR(p)$ is not empty, $p$ is one the convex hull". Which can easily be proved by contradiction.
However, the second part is a bit challenging for me, which is "If $p$ is on the convex hull, then $FVR(p)$ is not empty".
Can anyone help me on the proof for the second part?