The Fermat-Torricelli problem in the projective plane


  • Manolis C. Tsakiris
  • Sihang Xu



We pose and study the Fermat-Torricelli problem for a triangle in the projective plane under the sine distance. Our main finding is that if every side of the triangle has length greater than $\sin 60^\circ $, then the Fermat-Torricelli point is the vertex opposite the longest side. Our proof relies on a complete characterization of the equilateral case together with a deformation argument.


Afsari, B., Riemannian $L^p$ center of mass: existence, uniqueness, and convexity, Proc. Amer. Math. Soc. 139 (2011), no. 2, 655–673.

Arnaudon, M., and Miclo, L., Means in complete manifolds: uniqueness and approximation, ESAIM Probab. Stat. 18 (2014), 185–206.

Barbara, R., The Fermat-Torricelli points of n lines, Math. Gaz. 84 (2000), no. 499, 24–29.

Benítez, C., Fernández, M., and Soriano, M., Location of the Fermat-Torricelli medians of three points, Trans. Amer. Math. Soc. 354 (2002), no. 12, 5027–5038.

Benítez, C., and Yáñez, D., Middle points, medians and inner products, Proc. Amer. Math. Soc. 135 (2007), no. 6, 1725–1734.

Chakerian, G., and Ghandehari, M., The Fermat problem in Minkowski spaces, Geom. Dedicata 17 (1985), no. 3, 227–238.

Chen, X., Hardin, D. P., and Saff, E. B., On the search for tight frames of low coherence, J. Fourier Anal. Appl. 27 (2021), no. 1, Paper No. 2, 27 pp.

Cockayne, E., On the Steiner problem, Canad. Math. Bull. 10 (1967), no. 3, 431–450.

Cockayne, E., On Fermat's problem on the surface of a sphere, Math. Mag. 45 (1972), no. 4, 216–219.

Conway, J., Hardin, R., and Sloane, N., Packing lines, planes, etc.: Packings in Grassmannian spaces, Experiment. Math. 5 (1996), no. 2, 139–159.

Eriksson, F., The Fermat-Torricelli problem once more, Math. Gaz. 81 (1997), no. 490, 37–44.

Hajja, M., and Zachos, A., A complete analytical treatment of the weighted Fermat–Torricelli point for a triangle, J. Geom. 108 (2017), no. 1, 99–110.

Kupitz, Y. S., and Martini, H., Geometric aspects of the generalized Fermat-Torricelli problem, Bolyai Soc. Math. Stud. 6 (1997), 55–129.

Lin, B., and Yoshida, R., Tropical Fermat–Weber points, SIAM J. Discrete Math. 32 (2018), no. 2, 1229–1245.

Martini, H., Swanepoel, K., and Weiss, G., The Fermat–Torricelli problem in normed planes and spaces, J. Optim. Theory Appl. 115 (2002), no. 2, 283–314.

P., M. G., Solution des deux problèmes de trigonométrie sphérique énoncés à la pag. 64 du présent volume et de divers autres problèmes analogues, Ann. Math. Pures Appl. [Ann. Gergonne] 20 (1829), 137–151.

Taylor, D., Regular $2$-graphs, Proc. London Math. Soc. (3) 35 (1977), no. 2, 257–274.

Tsakiris, M. C., Dual principal component pursuit and filtrated algebraic subspace clustering, Ph.D. thesis, Johns Hopkins University, 2017.

Tsakiris, M. C., and Vidal, R., Hyperplane clustering via dual principal component pursuit, in International Conference on Machine Learning, 2017, pp. 3472–3481.

Uteshev, A., Analytical solution for the generalized Fermat-Torricelli problem, Amer. Math. Monthly 121 (2014), no. 4, 318–331.

Viviani, V., De maximis et minimis divinatio in quintum Conicorum Apollonii Pergaei. Part II, Florence 1659.

Yang, L., Riemannian median and its estimation, LMS J. Comput. Math. 13 (2010), 461–479.

Zachos, A. and Zouzoulas, G., The weighted Fermat-Torricelli problem and an“inverse”problem, J. Convex Anal. 15 (2008), no. 1, 55.



How to Cite

Tsakiris, M. C., & Xu, S. (2022). The Fermat-Torricelli problem in the projective plane. MATHEMATICA SCANDINAVICA, 128(3).