The Weiszfeld algorithm is an iterative algorithm to solve the Fermat-Weber problem. R. Chandrasekaran
and A. Tamir
[Math. Program., Ser. A 44, No. 3, 293–295 (1989; Zbl 0683.90026
)] stated the following conjecture: If the convex hull of the set of vertices is of full dimension, then the set of initial points for which the sequence generated by the Weiszfeld algorithm yields in a vertex is denumerable. J. Brimberg
[Math. Program. 71, No. 1 (A), 71–76 (1995; Zbl 0855.90075
)] claimed to prove the conjecture and extends it to a necessary and sufficient condition. The authors show in this paper that Brimberg’s proof is not correct. Moreover, they show by examples that the conjecture cannot be extended to a necessary and sufficient condition.