Is Minimax Linkage a Lance-Williams hierarchical clustering? I found the following article on "Hierarchical Clustering With Prototypes
via Minimax Linkage".

It is stated in Property 6 that 


  Minimax linkage cannot be written using Lance–Williams updates.


A succinct proof using a counter-example is given:


  Proof. 
  Figure 9 shows a simple one-dimensional example that could not
  arise if minimax linkage followed Lance– Williams updates. The upper
  and lower panels show two configurations of points for which the
  right side of (4) is identical but the left side differs; in
  particular, $d(G_1 \cup G_2,H) = 9$ for the upper panel, whereas
  $d(G_1 \cup G_2,H) = 8$ for the lower panel.


But I do not understand their proof. For both cases (upper and lower panels), $d(G_1,H) = 16$, $d(G_2,H) = 7$, $d(G_1,G_2) = 5$.

I cannot see any reason that $\alpha(G_2)$ in the first case must equal $\alpha(G_2)$ in the second case. For instance, $G_2$ has not the same cardinal.
My take after a quick read of the references.
First of all, Lance and Williams's original paper mentions that their linear scheme works (and offers computational advantage) only for combinatorial strategies. Is minimax linkage such a combinatorial strategy? In other words, does it depend (linearly) on pair-wise distances? By the defintion of minimax distance it is clear that this is implausible.
It is like the difference between mean and median computation in statistics. Mean is linear while Median is non-linear. There is no linear combination that can compute median from mean (although in certain limited cases they can coincide).
Second, the authors do not mention the form (or values) of the $\alpha, \beta, \gamma$ parameters in a hypothetical Lance-Williams method for minimax linkage. But in any case they are constants and $\alpha(\cdot)$ can be constant or rational functions of the respective cluster sizes (per the original Lance-Williams reference).
$G_2$ might not be of the same cardinality in the two panels, but minimax linkage distance depends on radius of cluster not cardinality (unlike average or centroid linkage) and the two examples have same radius thus $\alpha(G_2)$ is same in both cases.
Another way to see that this is the case is a plausible variation of the proof where $G_2$ in both cases has both same radius and same cardinality but different configuration.

Maybe such a proof would make it more clear. But I will leave it at that, at this point.