# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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.
##### MSC:
 90C35 Programming involving graphs or networks 00B55 Miscellaneous volumes of translations 00B60 Collections of reprinted articles 90C90 Applications of mathematical programming
##### References:
 [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. [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 . [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. [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). [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).