Proving this convex hull lemma

Question

I am trying to brush up on some computational geometry, and I am having trouble trying to prove the following:

given a planar point set P in general position, let k denote the number of hulls generated by the repeated hull process. There exists a point q in the plane such that every closed half plane whose bounding line l passing through q contains at least k points of P.

I got was looking at the questions from this PDF: https://www.cs.umd.edu/class/spring2012/cmsc754/Handouts/cmsc754-spring12-handouts.pdf

Problem 1c.

Any hints would be greatly appreciated.

Answer 1

Hint 1 : intuition (?)

When you're at point $q$, whatever direction you look you have $k$ points "in front" of you. The repeated hull process guarantees that you have $k$ "layers" of points in your point set $P$, so intuitively to have at least $k$ points in every direction, you want to put $q$ in the "innermost layer" of $P$.

Hint 2 : a result that can be useful (formal proof omitted)

If $q$ belongs to the convex hull of $P$, then any closed half-plane whose bounding line passes through $q$ contains at least one vertex of that convex hull.

Hint 3 : basically the solution

Take a point $q$ in the $k$-th convex hull ($H_k$ with the pdf's notation). Then $q$ belongs to every $H_i$, $1\le i\le k$.