The classification of finite connected hypermetric spaces

A finite distance space X , d d : X 2 --,7/ is hypermetric (of negative type) if axay d(x, y) < 0 for all integral sequences {axlx ~ X} that sum to 1 (sum to 0). X, d is connected if the set {(x, y)ld(x, y) = 1, x, y ~ X} is the edge set for a connected graph on X, and graphical if d is the path length distance for this graph. Then we prove Theorem 1. A connected space X, d has negative type if and only if X may be realised as a subset of a Euclidean space E, IJ II, sdch that (i) X contains 0 and spans E (ii} d(x,y) = 1/2tlx Yll 2 ( x , y ~ S ) (iii) L = Y_X is a root lattice, i.e. an orthogonal direct sum of lattices of type A,, D,, E6, ET, and Es. Call a hypermetric space X, d complete if for each triple x, y, z ~ X with d(y, z) = 1 and d(x, y) + 1 = d(x, z), there is a unique element w ~ X with d(w, x) = 1, d(w, y) = d(x, z), and d(w, z) = d(x, y). Then we also prove Theorem 2. (i) A connected distance space is hypermetric if and only if it is isomorphic to a subspace of a complete connected hypermetric spcae. (ii) The complete connected hypermetric spaces are graphical, and are precisely the Cartesian products of Johnson graphs, half cubes, Cocktail Party graphs, the Schlafli graph on 27 vertices, and the Gosset graph on 56 vertices. We finish by describing how a given connected hypermetric space may be canonically embedded in a complete one, and give some open problems. Theorem 1 is an extention of a result of Schoenberg. Theorem 2 is obtained by applying a result of Assouad to show any connected hypermetric space may be identified with a subset of a minimal saturated set induced by a coset of some root lattice in its lattice of weights.