Let us generalize this problem to r sets:
We have 3r + 2 points on the plane. We want to show that we can divide them into r sets of three points and one set of two points such that the intersection of the convex hulls of all r sets is not empty.
We take 3r+1 of the points and apply Tverberg’s theorem.
It gives us r disjoint nonempty sets which together contain all 3r+1 points. The intersection of the convex hulls of these sets is nonempty. Let on of the points in the intersection of the convex hulls be x.
If any of the four sets are bigger than 3 then by Caratheodory’s theorem there is a three point subset that contains x. We take that as the new set in place of the four point set and add the additional point to a set less than three. We repeat this process untill all of the sets are three or less. Then we add the remaining point to a set with two points or less. The result is 3r+2 points divided into r sets of three or less. The only way to do this is r sets of three points and one set of two points which gives the desired theorem.
We can add more points and keep the distribution even since adding a point to a set of points increases the convex hull so the intersection of convex hulls still contains x. So we have the following if we have 3r+2 points or more we can divide them into r nonempty sets, the size of each set differing by at most one element from each other set such that the intersection of the convex hulls of the sets is nonempty.
In the next of this series of posts I want to look at higher dimensional spaces than the plane.