On (super-)spherical distance matrices and two results from Schoenberg Original Research Article

A new characterization of simplicial distance matrices is obtained. The superspherical semidistance matrices are introduced. Two results deriving from Schoenberg are studied. A short proof of the first is obtained, before strengthening it to show that every circum-Euclidean semidistance matrix is superspherical. The second, that L2 embeds in L1, required a very sophisticated proof. The relationship between (super-)spherical and Euclidean matrices is clarified and, in the finite case, leads to an elementary proof of this second result. Finally, the classes of matrices studied here are located in the wider context of the classes studied by Critchley and Fichet (1994).