The stability of Fermat –Torricelli problem’s locus in normed planes
https://doi.org/10.22405/2226-8383-2025-26-2-90-100
Abstract
The article studies the structure of non-unique solutions of the Fermat–Torricelli problem in normed planes. The problem of the presence of the stability property for such solutions was posed. The results were obtained in the form of necessary and sufficient conditions for the
stability of all solutions for sets of three points in a normed plane. In addition, as an illustration, bifurcation diagrams of solutions were considered and their structure was investigated.
References
1. Bajaj, C., 1988, “The algebraic degree of geometric optimization problems”, Discr. Comput. Geom., vol. 3, pp. 177–191.
2. Boltyanski, V., Martini, H., Soltan, V., 1999, “Geometric methods and optimization problems”, Kluwer Acad. Publ.
3. Brazil, M., Graham, R.L., Thomas, D.A., Zachariasen, M., 2013, “On the History of the Euclidean Steiner Tree Problem”.
4. Cieslik, D., 1988, “The Fermat-Steiner-Weber-problem in Minkowski spaces”, Optimization, vol. 19, pp. 485–489.
5. Cockayne, E.J., Melzak, Z.A., 1969, “Euclidean constructibility in graph-minimization problems”, Math. Mag., vol. 42, pp. 206–208.
6. Durier, R., Michelot, C., 1985, “Geometrical properties of the Fermat-Weber problem”, Europ. J. Oper. Res., vol. 20, pp. 332–343.
7. Ilyukhin, D.A., 2022, “The Fermat–Torricelli problem in the case of three-point sets in normed planes”, Chebyshevskii Sbornik, vol. 23, no. 5, pp. 72–86.
8. Ivanov, A.O., Tuzhilin, A.A., 2002, “Branching geodesics in normed spaces”, Izvestiya: Mathematics, vol. 66, no. 5, pp. 905–948.
9. Ivanov, A.O., Tuzhilin, A.A., 2003, “Extreme Networks Theory”, Moscow-Izhevsk: Institute of Computer Investigations.
10. Kupitz, Y.S., Martini, H., 1997, “Geometric aspects of the generalized Fermat–Torricelli problem”, Bolyai Society Mathematical Studies, vol. 6, pp. 55–127.
11. Martini, H., Swanepoel, K.J., Weis, G., 2002, “The Fermat–Torricelli problem in normed planes and spaces”, Journal of Optimization Theory and Applications, vol. 115, pp. 283–314.
12. Nguyen, S.D., 2018, “Constrained Fermat-Torricelli-Weber Problem in real Hilbert Spaces”, ArXiv e-prints, arXiv:1806.04296.
13. Torricelli, E., 1919, “De maximis et minimis”, Opere di Evangelista Torricelli, Faenza, Italy.
14. Uteshev, A.Y., 2014, “Analytical solution for the generalized Fermat–Torricelli problem”, The American Mathematical Monthly, vol. 121, no. 4, pp. 318–331.
15. Zachos, A.N., 2014, “An analytical solution of the weighted Fermat-Torricelli problem on the unit sphere”, ArXiv e-prints, arXiv:1408.6495.
Review
For citations:
Ilyukhin D.A. The stability of Fermat –Torricelli problem’s locus in normed planes. Chebyshevskii Sbornik. 2025;26(2):90-100. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-2-90-100
Views:
13
Email this article 