A problem on distance matrices of subsets of the Hamming cube

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.