Measurement distances relations

Question

Suppose we have two clusters of points, $c_1, c_2$. We want to measure the distance between them. There are three popular ways:

1. single link = the short distance between $x\in c_1$ and $y\in c_2$
2. complete link = the longest distance between $x\in c_1$ and $y\in c_2$
3. group average link = $\frac{1}{|c_1\cup c_2|(|c_1\cup c_2|-1)}\sum_{x\in (c_1\cup c_2)}\sum_{y\in (c_1\cup c_2),y\ne x} dist(x,y)$

I'm trying to show $CL(1,2) \le (SL(1,2) + GAL(1,2)) / 2$

I'd be glad if you could help me with that

Thanks

Answer 1

Is the inequality true? Here's an idea for a counterexample.

Suppose that all the points of two large clusters are very close together. Then the shortest link and the average link will both be very small. Now add one point far away to the first cluster. That won't change the shortest link, and won't change the average link by much but can break the claimed inequality.