A problem on distance matrices of subsets of the Hamming cube

Abstract

Let D D denote the distance matrix for an n + 1 n+1 point metric space ( X , d ) (X,d) . In the case that X X is an unweighted metric tree, the sum of the entries in D − 1 D^{-1} is always equal to 2 / n 2/n . Such trees can be considered as affinely independent subsets of the Hamming cube H n H_n , and it was conjectured that the value 2 / n 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 H_n .