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On the point for which the sum of the distances to $n$ given points is minimum. (English) Zbl 1176.90616
This article presents a translation in English of a 1937 French article by E. Weiszfeld. The article begins with the statement of the theorem that is to be proven, which determines the properties of the point for which the sum of the distances to $n$ given points is a minimum. After the statement of the theorem, the article presents three different proofs. The original terminology is used throughout this translation. The article concludes with an annex containing notes and comments by the translator.

90C35Programming involving graphs or networks
00B55Miscellaneous volumes of translations
00B60Collections of reprinted articles
90C90Applications of mathematical programming
Full Text: DOI
[1] Cánovas, L., Canavate, R., & Marín, A. (2002). On the convergence of the Weiszfeld algorithm. Mathematical Programming, 93, 327--330. · Zbl 1065.90054 · doi:10.1007/s101070200297
[2] Drezner, Z., Klamroth, K., Schöbel, A., & Wesolowski, G. O. (2002). The Weber problem. In Z. Drezner & H. Hamacher (Eds.), Facility location: applications and theory (pp. 1--36). Berlin: Springer. · Zbl 1041.90023
[3] Franksen, O. I., & Grattan-Guinness, I. (1989). The earliest contribution to location theory? Spatio-temporal equilibrium with Lamé and Clapeyron, 1829. Mathematics and Computers in Simulation, 31, 195--220. · Zbl 0672.01021 · doi:10.1016/0378-4754(89)90159-6
[4] Gass, S. A. (2004). In Memoriam, Andrew (Andy) Vazsonyi: 1916--2003. OR/MS Today, February 2004. http://www.lionhrtpub.com/orms/orms-2-04/frmemoriam.html , see also this volume.
[5] Hardy, G. H. (1940). A mathematician’s apology. London, now freely available at http://www.math.ualberta.ca/$\sim$mss/books/AMathematician’sApology.pdf . · Zbl 0025.19301
[6] Kuhn, H. W., & Kuenne, R. E. (1962). An efficient algorithm for the numerical solution of the generalized Weber problem in spatial economics. Journal of Regional Science, 4, 21--33. · doi:10.1111/j.1467-9787.1962.tb00902.x
[7] Kupitz, Y. S., & Martini, H. (1997). Geometric aspects of the generalized Fermat-Torricelli problem. In Mathematical studies: Vol. 6. Intuitive geometry (pp. 55--127). Bolyai Society. · Zbl 0911.51021
[8] Lamé, G., & Clapeyron, B. P. E. (1829). Mémoire sur l’application de la statique à la solution des problèmes relatifs à la théorie des moindres distances. Journal des Voies de Communications, 10, 26--49. (In french--Memoir on the application of statics to the solution of problems concerning the theory of least distances.) For a translation into English see Franksen, O.I., Grattan-Guinness, I. (1989). Mathematics and Computers in Simulation, 31, 195--220.
[9] Sturm, R. (1884). Ueber den Punkt kleinster Entfernungssumme von gegebenen Punkten. Journal für die reine und angewandte Mathematik, 97, 49--61. (In german--On the point of smallest distance sum from given points). · doi:10.1515/crll.1884.97.49
[10] Vazsonyi, A. (2002a). Which door has the Cadillac. New York: Writers Club Press.
[11] Vazsonyi, A. (2002b). Pure mathematics and the Weiszfeld algorithm. Decision Line, 33(3), 12--13. http://www.decisionsciences.org/DecisionLine/Vol33/33_3/index.htm .
[12] Weiszfeld, E. (1936). Sur un problème de minimum dans l’espace. Tôhoku Mathematical Journal, 42, 274--280. (First series). · Zbl 62.0711.03
[13] Weiszfeld, E. (1937). Sur le point pour lequel la somme des distances de n points donnés est minimum. Tôhoku Mathematical Journal, 43, 355--386. (First series). · Zbl 63.0583.01