  Let $D$ denote the distance matrix for an $n+1$ point metric space $(X,d)$.
In the case that $X$ is an unweighted metric tree, the sum of the entries in
$D^{-1}$ is always equal to $2/n$. Such trees can be considered as affinely
independent subsets of the Hamming cube $H_n$, and it was conjectured that the
value $2/n$ was minimal among all such subsets. In this paper we confirm this
conjecture and give a geometric interpretation of our result which applies to
any subset of $H_n$.
