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The Fermat-Weber location problem revisited. (English) Zbl 0855.90075
Summary: The Fermat-Weber location problem requires finding a point in ${ℝ}^{N}$ that minimizes the sum of weighted Euclidean distances to $m$ given points. A one-point iterative method was first introduced by Weiszfeld in 1937 to solve this problem. Since then several research articles have been published on the method and generalizations thereof. Global convergence of Weiszfeld’s algorithm was proven in a seminal paper by Kuhn in 1973. However, since the $m$ given points are singular points of the iteration functions, convergence is conditional on none of the iterates coinciding with one of the given points. In addressing this problem, Kuhn concluded that whenever the $m$ given points are not collinear, Weiszfeld’s algorithm will converge to the unique optimal solution except for a denumerable set of starting points. As late as 1989, Chandrasekaran and Tamir demonstrated with counter-examples that convergence may not occur for continuous sets of starting points when the given points are contained in an affine subspace of ${ℝ}^{N}$. We resolve this open question by proving that Weiszfeld’s algorithm converges to the unique optimal solution for all but a denumerable set of starting points if, and only if, the convex hull of the given points is of dimension $N$.

##### MSC:
 90B85 Continuous location