A problem on distance matrices of subsets 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 $H_n$.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.07052
 Bibcode:
 2021arXiv210907052D
 Keywords:

 Mathematics  Metric Geometry;
 Mathematics  Functional Analysis;
 Primary 46B85;
 Secondary 15A45;
 51K99
 EPrint:
 12 pages