A simple but curious statement:
In any finite set of points on the plane, there are points that have less than 4 closest neighbors.
Not sure if this is a deep or useful observation.
Not sure if I have a rigorous proof. I’ll publish a sketch of a proof later.
And here is how you prove it:
Let s be the minimum pairwise distance in our set. Let S be the subset containing all points that have neighbors at the distance s.
Consider C=ConvexHull(S). There is at least one vertex in C with the angle less than 180 degrees. Easy to show that that vertex has at most 3 closest neighbors.
Ok, so there is no set in which each point has 4 (or more) closest neighbors.
Now build a set in which each point has exactly 3 closest neighbors. What is the minimal number of points in such set?