On yielding and jointly yielding entries of Euclidean distance matrices

An $n\times n$ matrix D is a Euclidean distance matrix (EDM) if there exist $p^1,\dots ,p^n$ in some Euclidean space such that $d_{ij} = || p^i - p^j||^2$forall$i,j=1,\ldots,n$\mathrm{.}LetDbeanEDMandlet$E^{ij}$bethe$n \times n$symmetricmatrixwith{1}^{\mathrm{\prime }}sinthe$ij$thand$ji$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 $l_{ij} \leq t \leq u_{ij}$\mathrm{.}Iftheyieldingintervalofentry$d_{ij}$haslength0,i\mathrm{.}e\mathrm{.},if$l_{ij}=u_{ij}$,then$d_{ij}$issaidtobeunyielding\mathrm{.}Otherwise,if$l_{ij} \neq u_{ij}$,then$d_{ij}$issaidtobeyielding\mathrm{.}Let$d_{ij}$and$d_{ik}$betwounyieldingentriesof$D$\mathrm{.}Wesaythat$d_{ij}$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 $p^1,\dots ,p^n$. Moreover, for each yielding entry, we present explicit formulae of its yielding interval. Finally, we specialize our results to the case where $p^1,\dots ,p^n$ are in general position.