Circum-Euclidean distance matrices and faces Original Research Article

We study the structure of circum-Euclidean distance matrices, those Euclidean distance matrices generated by points lying on a hypersphere. We show, for example, that such Euclidean distance matrices are characterized as having constant row sums and they constitute the interior of the cone of all Euclidean distance matrices. Also, we provide a formula for computing the radius of a representing configuration in the smallest embedding dimension r and show that rk D = r + 1. Finally we obtain a geometric characterization of the faces of this cone. Given a configuration of points and its Euclidean distance matrix D, any matrix in the minimal face containing D comes from a configuration that is a linear perturbation of the points that generate D.