On the geometric median and other median-like points

Occupation: Professor,
Affiliation:
Central Economics and Mathematics Institute, Russian Academy of Sciences
Adygeya StateUniversity
Dmitry Pozharsky University
Moscow Institute of Physics and Technology (State University)

Address: Russia

Journal name

Ekonomika i matematicheskie metody

Edition

Volume 56 Issue 3

Pages

145-152

Abstract

The geometric median and some of its generalizations are widely used in economic theory, starting with the works of Wilhelm Launhardt and Alfred Weber on location theory.The most important property of the median of a numerical sample is that the median minimizes the sum of the distance to all elements of the sample.This minimizing property is the basis for determining the geometric median for finite sets of points on the plane.This definition is easily transferred to any metric space, including the Euclidean space R^n. Using integration, the concept of a geometric median extends to bounded submanifolds of any dimension inRⁿ.There are effective numerical methods for finding the geometric median, but there are no General analytical formulas for calculating it. In this paper, we focus on the geometric medians of bounded domains located in the Euclidean space Rⁿ.The main new results obtained in our work include the conclusion of a new convenient representation of the gradient system for finding the geometric median, as well as the extension of this approach to a wide class of similar optimization problems, where the distance function is replaced by functions of a more general form. It is the solutions to these problems that we call median-like points. They are the closest relatives of the geometric median and are also widely used in modern economic studies.

Keywords

geometric median, domain, triangular domain, gradient system, critical point.

Received

10.07.2020

Publication date

04.09.2020

Number of characters

18004

Cite

GOST	Panov P., Savvateev A. On the geometric median and other median-like points // Ekonomika i matematicheskie metody – 2020. – V. 56. – Issue 3 C. 145-152 [Electronic resource]. URL: https://emm.jes.su/S042473880010498-6-1 (circulation date: 16.04.2024). DOI: 10.31857/S042473880010498-6
MLA	Panov, Peter, Savvateev, Aleksei "On the geometric median and other median-like points." Ekonomika i matematicheskie metody 56.3 (2020):145-152. DOI: 10.31857/S042473880010498-6
APA	Panov P., Savvateev A. (2020). On the geometric median and other median-like points. Ekonomika i matematicheskie metody 56(3), pp.145-152 DOI: 10.31857/S042473880010498-6

100 rub.

When subscribing to an article or issue, the user can download PDF, evaluate the publication or contact the author. Need to register.

Number of purchasers: 0, views: 710

Readers community rating: votes 0

1. Divergence theorem (2014). Encyclopedia of Mathematics. Available at: http://www.encyclopediaofmath.org/index.php?title=Divergence_theorem&oldid=31341

2. Fekete S., Mitchell J., Beurer K. (2005). On the continuous Fermat Weber problem. Oper. Res., 53, 1, 6176.

3. FermatTorricelli problem (2012). Encyclopedia of Mathematics. Available at: https://www.encyclopediaofmath.org/index.php/Fermat-Torricelli_problem

4. Kemperman J.H.B. (1987). The median of a finite measure on a Banach space. In: Statistical data analysis based on the L1-norm and related methods. Amsterdam: North-Holland, 217230.

5. Kimberling C. (2020). Encyclopedia of triangle centers. Available at: http://faculty.evansville.edu/ck6/encyclopedia/

6. Launhardt W. (1882). Die Bestimmung des zweckmassigsten Standortes einer gewerblichen An-lage. Zeitschrift des Vereines deutscher Ingenieure, 26, 106115.

7. Panov P.A. (2017). Nash equilibria in the facility location problem with externalities. Journal of the New Economic Association, 1 (33), 2842 (in Russian).

8. Panov P.A. (2018). On the geometric median of convex, triangular and other polygonal domains. The Bulletin of Irkutsk State University. Series Mathematics, 26, 6275 (in Russian).

9. Savvateev A., Sorokin C., Weber S.? (2015).? Multidimensional? free-mobility? equilibrium: Tiebout ?revisited.?Preprint.

10. Weber A. (1909). Uber den Standort der Industrien. Erster Teil. Reine Theorie des Standorts. Ver-lag von J.C.B. Mohr (Paul Siebeck). Tubingen.

11. Weber problem (2014) Encyclopedia of Mathematics. Available at: https://www.encyclopediaofmath.org/index.php/Weber_problem

12. Wesolowsky G.O. (1993). The Weber problem: History and perspectives. Location Sciense, 1, P. 523.

13. Zhang T., Carlsson J. (2014). On the Continuous Fermat Weber Problem for a Convex Polygon Using Euclidean Distance, http://arxiv.org/abs/1403.3715