In the Fermat-Weber problem, the location of a source point in ℝN is sought which minimizes the sum of weighted Euclidean distances to a set of destinations. A classical iterative algorithm known as the Weizsfeld procedure is used to find the optimal location. Kuhn proves global convergence except for a denumerable set of starting points, while Katz provides local convergence results for this algorithm. In this paper, the authors consider a generalized version of the Fermat-Weber problem, where distances are measured by an lp norm and the parameter p takes on a value in the closed interval. This permits the choice of a continuum of distance measures from rectangular (p=1) to Euclidean (p=2). An extended version of the Weiszfeld procedure is presented and local convergence results obtained for the generalized problem. Linear asymptotic convergence rates are typically observed. However, in special cases where the optimal solution occurs at a singular point of the iteration functions, this rate can vary from sublinear to quadratic. It is also shown that for sufficiently large values of p exceeding 2, convergence of the Weiszfeld algorithm will not occur in general.
