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The Fermat–Torricelli theorem in convex geometry

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Abstract

This paper studies a generalization of the Euclidean triangle, the generalized deltoid, which we believe to be the right one for convex geometry. To illustrate the process, our main result shows that the generalized deltoid satisfies a convex generalization of the Fermat–Torricelli theorem. A point that minimizes the sum of distances to the vertices of a triangle (Fermat–Torricelli point) is the same as one through which pass three equiangular affine diameters (Fermat–Ceder point). A generalized deltoid is a triangle whose sides are disjoint, outwardly-looking arcs of convex curves. The Fermat–Torricelli theorem in convex geometry extends the Fermat–Ceder point of a triangle to a Fermat–Ceder point of a generalized deltoid. As an application, we show that the Fermat–Ceder points for the continuous families of affine diameters, area-bisecting lines, and perimeter-bisecting lines are unique for every triangle, and non-unique for every pentagon. In the case of quadrilaterals, the uniqueness of the Fermat–Ceder point for affine diameters holds precisely for all non-trapezoids, the one for the Fermat–Ceder point for area-bisecting lines holds for all quadrilaterals, and the one for the Fermat–Ceder point for perimeter-bisecting lines is open.

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Acknowledgements

The authors would like to thank the anonymous referee for his careful reading and suggestions for improvement. We are grateful to Horst Martini for his proof reading of an earlier version of the manuscript, his generosity and valuable insight. In addition, we thank Marilyn Breen, Paul Goodey, Rolf Schneider and Tudor Zamfirescu for their support.

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    Allan Berele & Stefan Catoiu

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  1. Allan Berele
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Correspondence to Stefan Catoiu.

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Berele, A., Catoiu, S. The Fermat–Torricelli theorem in convex geometry. J. Geom. 111, 22 (2020). https://doi.org/10.1007/s00022-020-00535-6

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