The hypermetric cone on $8$ vertices and some generalizations

The lists of facets -- $298,592$ in $86$ orbits -- and of extreme rays -- $242,695,427$ in $9,003$ orbits -- of the hypermetric cone $HYP_8$ are computed. The first generalization considered is the hypermetric polytope $HYPP_n$ for which we give general algorithms and a description for $n\le 8$. Then we shortly consider generalizations to simplices of volume higher than $1$, hypermetric on graphs and infinite dimensional hypermetrics.