The Hypermetric Cone on Seven Vertices

Abstract

The hypermetric cone HYP_n is the set of vectors (d_ij )_1≤i<i≤n satisfying the inequalities

A Delaunay polytope of a lattice is called extreme if the only affine bijective transformations of it into a Delaunay polytope, are the homotheties; there is correspondence between such Delaunay polytopes and extreme rays of HYP_n . We show that unique Delaunay polytopes of root lattices A _l and E ₆ are the only extreme Delaunay polytopes of dimension at most 6. We describe also the skeletons and adjacency properties of HYP ₇ and of its dual.

The computational technique used is polyhedral, i.e., enumeration of extreme rays, using the program cdd [Fukuda 03], and groups to reduce the size of the computations.

Keywords:

• Cone
• lattice
• hypermetric
• Delaunay polytope