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Cones of Weighted and Partial Metrics

A partial semimetric on V_n={1, ..., n} is a function f=((f_{ij})): V_n^2 -> R_>=0 satisfying f_ij=f_ji >= f_ii and f_ij+f_ik-f_jk-f_ii >= 0 for all i,j,k in V_n. The function f is a weak partial semimetric if f_ij >= f_ii is dropped, and it is a strong partial semimetric if f_ij >= f_ii is complemented by f_ij <= f_ii+f_jj. We describe the cones of weak and strong partial semimetrics via corresponding weighted semimetrics and list their 0,1-valued elements, identifying when they belong to extreme rays. We consider also related cones, including those of partial hypermetrics, weighted hypermetrics, l_1-quasi semimetrics and weighted/partial cuts.