The Hypermetric Cone on Seven Vertices

Abstract

The hypermetric cone {\small $HYP_n$} is the set of vectors {\small $(d_{ij})_{1\leq i< j\leq n}$} satisfying the inequalities

\sum_{1\leq i<j\leq n} b_ib_jd_{ij}\leq 0\mbox{with}b_i\in\Z\mbox{and}\sum_{i=1}^{n}b_i=1\,.

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 {\small $HYP_n$}. We show that unique Delaunay polytopes of root lattices {\small $A_1$} and {\small $E_6$} are the only extreme Delaunay polytopes of dimension at most 6. We describe also the skeletons and adjacency properties of {\small $HYP_7$} 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.