A problem on distance matrices of subsets of the Hamming cube

TL;DR

This paper proves a conjecture that the sum of entries in the inverse of the distance matrix for certain subsets of the Hamming cube is minimized at a specific value, with a geometric interpretation applicable to all subsets.

Contribution

It confirms the conjecture that the sum is minimized at 2/n for subsets of the Hamming cube and provides a geometric interpretation of this result.

Findings

Sum of entries in D^{-1} equals 2/n for unweighted metric trees.

The minimal sum 2/n is achieved by affinely independent subsets of the Hamming cube.

The result applies broadly to any subset of the Hamming cube.

Abstract

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 $H_n$.