Circum-Euclidean distance matrices and faces
A nonnegative symmetric \(n \times n\) matrix \(D = (d_{ij}^2)\) with zeros on the diagonal is called a dissimilarity matrix. If there exist \(n\) points \(p_1, \dots, p_n\) in \(\mathbb{R}^r\) such that \(d_{ij}^2 = |p_i - p_j |^2\), then \(D\) is called Euclidean distance matrix where \(|p - q |\) denotes the usual Euclidean distance. The smallest such value of \(r\) is called the embedding dimension. \(D\) is called a circum-Euclidean provided the points which generate it lie on the surface of some hypersphere. The authors study the structure of circum-Euclidean distance matrices, they show 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. The authors 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, they present 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 \(D\).
scientific article; zbMATH DE number 841755
scientific article