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 p, . . . , 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+ tE ij 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 6= 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 + t1E ij + t2E ik 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 transform of p, . . . , pn. Moreover, for each yielding entry, we present explicit formulae of its yielding interval. Finally, we specialize our results to the case where p, . . . , pn are in general position.