  The hypermetric cone $HYP_n$ is the set of vectors $(d_{ij})_{1\leq i< j\leq
n}$ satisfying the inequalities $\sum_{1\leq i<j\leq n} b_ib_jd_{ij}\leq 0 with
b_i\in\Z and \sum_{i=1}^{n}b_i=1$. 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 $HYP_n$. 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
$HYP_7$ and of its dual.
