Translation with annotations of E. Weiszfeld, Sur le point pour lequel la somme des distances de n points donnés est minimum, Tôhoku Mathematical Journal (first series), 43 (1937) pp. 355-386.A short introduction about the translation is found in Appendix A. Appendix B lists particular notations used by Weiszfeld and their now more conventional equivalents. Numbered footnotes are those of the original paper of Weiszfeld. Boxed numerals are references to observations about the translation and comments of the translator, all to be found in Appendix C.Keywords Location problem · Weber problem · Euclidean distance · Weiszfeld algorithmThe following theorem is known 1 :Theorem Let n different points A 1 , A 2 , . . . , A n not lying on a straight line be given in space. There then exists a single point for which the sum of distances from these n points is minimum. For this point M we then have the inequalitywhich is valid for any point X different from M. (There can, of course, exist at most one point of this nature.) This point, which we shall call minimum-point, is characterized by the fact that 1 R. Sturm, Journal f. Mathematik, 97.
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