# On the point for which the sum of the distances to n given points is minimum

@article{Vazsonyi2009OnTP, title={On the point for which the sum of the distances to n given points is minimum}, author={Andrew Vazsonyi and Frank Plastria}, journal={Annals of Operations Research}, year={2009}, volume={167}, pages={7-41} }

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…

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## References

SHOWING 1-10 OF 15 REFERENCES

On the convergence of the Weiszfeld algorithm

- Mathematics, Computer ScienceMath. Program.
- 2002

This work analyzes the paper “Brimberg, J. (1995): The Fermat-Weber location problem revisited” which claims to close the question on the conjecture posed by Chandrasekaran and Tamir in 1989 on the convergence of the Weiszfeld algorithm.

A mathematician's apology

- Art, Computer Science
- 1940

Reading online A Mathematician's Apology book will be provide using wonderful book reader and it's might gives you some access to identifying the book content before you download the book.

Facility location - applications and theory

- Engineering, Computer Science
- 2001

The Weber Problem, a study of location problems in the public sector, and an Efficient Genetic Algorithm for the p-Median Problem, provide insights into the design of location models.

The earliest contribution to location theory? Spatio-economic equilibrium with Lame´ and Clapeyron, 1829

- Economics
- 1989

Pure mathematics and the Weiszfeld algorithm

- Decision Line, 33(3), 12–13. http://www.decisionsciences.org/DecisionLine/Vol33/33_3/index.htm.
- 2002

Which door has the Cadillac

- New York: Writers Club Press.
- 2002

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.
- 2004

Geometric aspects of the generalized Fermat-Torricelli problem

- In Mathematical studies:
- 1997