Delaunay Polytopes: Rank and Hypermetric Faces

Abstract

There is a natural notion of rank for hypermetric spaces. Namely, if (X, d) is a hypermetric space, then its rank rk(X,d) is defined as the dimension of the smallest face of the cone HYP(X) that contains d. The extremal cases when the rank of (X, d) or its corank is equal to 1 correspond, respectively, to the cases when d lies on an extreme ray or on a facet of HYP(X). Correspondingly, the rank rk(P) of a Delaunay polytope P is defined as the rank of its Delaunay polytope space (V(P),d ⁽²⁾). Delaunay polytopes of rank 1 are called extreme; they are associated to hypermetrics lying on an extreme ray of the hypermetric cone. This notion of rank for a Delaunay polytope P has the following geometric interpretation: It coincides with the number of degrees of freedom one has when deforming P in such a way that the deformed polytope remains a Delaunay polytope; a precise formulation can be found in Theorem 15.2.5.