On the Fermat—Weber problem with convex cost functions

We treat an extension of the generalized Fermat—Weber problem with convex cost functions. It is shown that the entire sequence of iterates (as opposed to selected subsequences) generated by each of the two proposed algorithms converges to a minimum although the economic function is not strictly convex. The general idea is to associate, with the economic function calledh, a family of more regular strictly convex functions, the lower envelope of which is the functionh.