On yielding and jointly yielding entries of Euclidean distance matrices

Abstract

An n×n matrix D is a Euclidean distance matrix (EDM) if there exist points p1,…,pn in some Euclidean space such that dij=‖pi−pj‖2 for all i,j=1,…,n. Let D be an EDM and let Eij be the n×n symmetric matrix with 1's in the ijth and jith entries and 0's elsewhere. We say that [lij,uij] is the yielding interval of entry dij if it holds that D+tEij is an EDM if and only if lij≤t≤uij. If the yielding interval of entry dij has length 0, i.e., if lij=uij=0, then dij is said to be unyielding. Otherwise, if lij≠uij, then dij is said to be yielding. Let dij and dik be two unyielding entries of D. We say that dij and dik are jointly yielding if D+t1Eij+t2Eik is an EDM for some nonzero scalars t1 and t2.In this paper, we characterize the yielding and the jointly yielding entries of an EDM D in terms of Gale transforms of p1,…,pn. Moreover, for each yielding entry, we present explicit formulae of its yielding interval. Finally, we specialize our results to the case where points p1,…,pn are in general position.