Cones of weighted quasi-metrics, weighted quasi-hypermetrics and of   oriented cuts
  We show that the cone of weighted n-point quasi-metrics WQMet_n, the cone of weighted quasi-hypermetrics WHyp_n and the cone of oriented cuts OCut_n are projec- tions along an extreme ray of the metric cone Metn+1, of the hypermetric cone Hypn+1 and of the cut cone Cut_{n+1}, respectively. This projection is such that if one knows all faces of an original cone then one knows all faces of the projected cone. 
Michel Deza, Vyacheslav Grishukhin and Elena Deza
math.MG