Exercise 4.7.8

We define the minimum distance $D^{\text{∗}}$ between two clusters to be the minimum distance among all distances between the points from cluster $P$ to points in cluster $Q$.

Suppose we separate all the points into two clusters $P,Q$, By observation, the maximum of the minimum distance can be achieved by either: * Put point $(0,4)$ into one cluster * Put point $(0,8)$ into one cluster * Put points $(0,4),(0,8)$ into one cluster

All three ways of clustering have the minimum distance $D^{\text{∗}}=4$, which is the largest among all possible clusterings.