Cones of partial metrics

A partial semimetric on a set X is a function $(x, y) mapsto p(x, y) in RR_{geq 0}$ satisfying $p(x, y) = p(y, x)$, $p(x, y) geq p(x, x)$ and $p(x, z) leq p(x, y) + p(y, z) p(y, y)$ for all $x, y, z in X$. We study here the polyhedral convex cone $PSMET_n$ of all partial semimetrics on $n$ points, using computations done for $n leq 6$. We present data on those cones and their relatives: the number of facets, of extreme rays, of their orbits, incidences, characterize ${0, 1}$- valued extreme rays, etc.