The hypermetric cone on seven vertices

Abstract

The hypermetric cone $HY{P}_n$ is the set of vectors $(d_{ij}{)}_{1\le i<j\le n}$ satisfying the inequalities ${\sum }_{1\le i<j\le n}{b}_i{b}_j{d}_{ij}\le 0with{b}_i\in \mathbb{Z}and{\sum }_{i=1}^n{b}_i=1$. A Delaunay polytope of a lattice is called extremal if the only affine bijective transformations of it into a Delaunay polytope, are the homotheties; there is a correspondance between such Delaunay polytopes and extreme rays of $HY{P}_n$. We show that unique Delaunay polytopes of root lattice $A_1$ and $E_6$ are the only extreme Delaunay polytopes of dimension at most 6. We describe also the skeletons and adjacency properties of $HY{P}_7$ and of its dual.

8 pages, 4 tables