Clifford’s Identity and Generalized Cayley-Menger Determinants

Distance geometry is usually defined as the characterization and study of point sets in \({\mathbb R}^k\), the k-dimensional Euclidean space, based on the pairwise distances between their points. In this paper, we use Clifford’s identity to extend this kind of characterization to sets of n hyperspheres embedded in \({\mathbb S}^{n-3}\) or \({\mathbb R}^{n-3}\) where the role of the Euclidean distance between two points is replaced by the so-called power between two hyperspheres. By properly choosing the value of n and the radii of these hyperspheres, Clifford’s identity reduces to conditions in terms of generalized Cayley-Menger determinants which has been previously obtained on the basis of a case-by-case analysis.