CIRCUM-EUCLIDEAN DISTANCE MATRICES AND FACES

We study the structure of circum-Euclidean distance matrices, those Eu clidean distance matrices generated by points lying on a hypersphere. We show, for example, that such Euclidean distance matrices are charac terized as having constant row sums and they constitute the interior o f the cone of all Euclidean distance matrices. Also, we provide a form ula for computing the radius of a representing configuration in the sm allest embedding dimension r and show that rk D = r + 1. Finally we ob tain a geometric characterization of the faces of this cone. Given a c onfiguration 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.