Faces of the cone of Euclidean distance matrices: Characterizations, structure and induced geometry

In this paper we obtain new characterizations of the faces of the cone of Euclidean distance matrices. In one case the characterization is based on a special subspace associated with each distance matrix, in other case on linear restrictions of coordinate matrices. We also relate the faces to the supporting hyperplanes of the cone, and we show how the supporting hyperplanes induce certain geometry on the configurations that belong to that face.