# Prove that for any n > 3 there is a set of n point sites in the plane such that one of the cells of Voronoi diagram(P) has n − 1 vertices

I am trying to solve some exercises of the book "Computational Geometry Algorithm and Applications, 3rd - de berg et al" of chapter 7 - Voronoi Diagrams. Unfortunately, I am not sure if I understand the following question:

Prove that for any n > 3 there is a set of n point sites in the plane such that one of the cells of Voronoi diagram(P) has n − 1 vertices

The problem is when I look at the following figure of the book: There are 11 site points, but I do not see any cell with 10(11-1) vertices. Does this mean I have to proof that this is not true? Or did I not understand the question correctly?

I hope that someone could show me how to solve this question.

I am not sure if this is allowed, but I also posted my question here: https://stackoverflow.com/questions/36208503/prove-that-for-any-n-3-there-is-a-set-of-n-point-sites-in-the-plane-such-that The reason that I posted the same question on stackoverflow and math.stackexchange is that someone in the comments told me that math.stackexchange is a better place for this question.

## 1 Answer

You did not understand the question correctly: The question says to show that there is SOME configuration with this property, not that every configuration has the property.

Consider the following statement: "For any open interval in the real line, there's a pair of points whose distance is rational. "

You say: In the interval $-1 < x < 1$, the points $-\pi/4$ and $\pi/4$ have a non-rational distance.

That's true, but it doesn't falsify the claim, because the points $\pm \frac{1}{2}$ have a rational distance.