Extreme Delaunay Polytopes

Abstract

In this chapter, we consider extreme Delaunay polytopes, i.e., Delaunay poly-topes with rank 1. A geometric characterization of extreme Delaunay polytopes has been given in Corollary 15.2.4. Extreme Delaunay polytopes are of particular interest since they correspond to the extreme rays of the hypermetric cone. More precisely, if d ∈ HYP n lies on an extreme ray of HYP n , then its associated Delaunay polytope P d is an extreme Delaunay polytope of dimension k≤ n − 1. Conversely, if P is a k-dimensional extreme Delaunay polytope then, for each generating subset V of its set of vertices, the hypermetric space (V, d (2)) lies on an extreme ray of the hypermetric cone HYP(V). Moreover, by taking gate 0-extensions of (V, d (2)), we obtain extreme rays of the cone HYP n for any n ≥ |V|. In particular, if P is basic, then each basic subset of V(P) yields an extreme ray of the hypermetric cone HYP k + 1l and, thus, of HYP n for n ≥ k + 1. Therefore, finding all extreme rays of the hypermetric cone HYP n yields the question of finding all extreme Delaunay polytopes of dimension k ≤ n − 1.