On yielding and jointly yielding entries of Euclidean distance matrices

TL;DR

This paper characterizes the yielding and jointly yielding entries of Euclidean distance matrices using Gale transforms, providing explicit formulas for yielding intervals and analyzing the case of points in general position.

Contribution

It introduces a characterization of yielding and jointly yielding entries of EDMs via Gale transforms and derives explicit formulas for yielding intervals.

Findings

Characterization of yielding entries using Gale transforms.

Explicit formulas for yielding intervals.

Analysis of the case with points in general position.

Abstract

An $n \times n$ matrix D is a Euclidean distance matrix (EDM) if there exist $p^1, \ldots, p^n$ in some Euclidean space such that $d_{ij} = || p^i - p^j||^2$ for all $i,j=1,\ldots,n$. Let D be an EDM and let $E^{ij}$ be the $n \times n$ symmetric matrix with 1's in the $ij$th and $ji$th entries and 0's elsewhere. We say that $[l_{ij},u_{ij}]$ is the yielding interval of entry $d_{ij}$ if it holds that $D+t E^{ij}$ is an EDM iff $l_{ij} \leq t \leq u_{ij}$. If the yielding interval of entry $d_{ij}$ has length 0, i.e., if $l_{ij}=u_{ij}$, then $d_{ij}$ is said to be unyielding. Otherwise, if $l_{ij} \neq u_{ij}$, then $d_{ij}$ is said to be yielding. Let $d_{ij}$ and $d_{ik}$ be two unyielding entries of $D$. We say that $d_{ij}$ and $d_{ik}$ are jointly yielding if $D+t_1 E^{ij} + t_2 E^{ik}$ is an EDM for some nonzero scalars $t_1$ and $t_2$. In this paper, we characterize the yielding and the jointly yielding entries of an EDM D in terms of Gale transform of $p^1,\ldots,p^n$. Moreover, for each yielding entry, we present explicit formulae of its yielding interval. Finally, we specialize our results to the case where $p^1,\ldots,p^n$ are in general position.