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arXiv · 2109.07052

A problem on distance matrices of subsets of the Hamming cube

Ian Doust·Reinhard Wolf
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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 $H_n$.

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Ian Doust, Reinhard Wolf. 2021-09-15. A problem on distance matrices of subsets of the Hamming cube. https://doi.org/10.1090/bproc%2F122

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