ON (SUPER-)SPHERICAL DISTANCE MATRICES AND 2 RESULTS FROM SCHOENBERG

A new characterization of simplicial distance matrices is obtained. Th e superspherical semidistance matrices are introduced. Two results der iving from Schoenberg are studied. A short proof of the first is obtai ned, before strengthening it to show that every circum-Euclidean semid istance matrix is superspherical. The second, that L(2) embeds in L(1) , required a very sophisticated proof. The relationship between(super- ) spherical and Euclidean matrices is clarified and, in the finite cas e, leads to an elementary proof of this second result. Finally, the cl asses of matrices studied here are located in the wider context of the classes studied by Critchley and Fichet (1994). (C) Elsevier Science Inc., 1997