The hypermetric cone on seven vertices

Original · EN

The hypermetric cone HYPₙ is the set of vectors (dij)₁≤ ᵢ< ⱼ≤ ₙ satisfying the inequalities ∑₁≤ ᵢ<ⱼ≤ ₙ bᵢbⱼdij≤ 0 with bᵢ∈ and ∑ᵢ₌₁ⁿbᵢ=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 HYPₙ. We show that unique Delaunay polytopes of root lattice A₁ and E₆ are the only extreme Delaunay polytopes of dimension at most 6. We describe also the skeletons and adjacency properties of HYP₇ and of its dual.