Maximizers and Minimizers of Functions Defined by the Distances From the Edges of Spherical and Hyperbolic Triangles

Abstract

In this paper, we investigate the minimizers and maximizers of functions defined by the sum of the distances from the edges of a given triangle by extending it to the setting of the sphere and the hyperbolic plane. We also study the asymptotic behavior of these points as the curvature of the space to zero.

Keywords

• Triangle
• Sphere
• Hyperbolic plane
• Distances from edges

References

1. [1] E. Abboud, Minimal sum of powered distances from the sides of a triangle, arXiv:1707.01853 [math.OC], 7pages. $ \href{https://doi.org/10.48550/arXiv.1707.01853}{\mbox{[CrossRef]}} $
2. [2] C. Kimberling, Trilinear distance inequalities for the symmedian point, the centroid, and other triangle centers, Forum Geom., 10 (2010), 135-139.
3. [3] J. Casey, A Treatise on Spherical Trigonometry, and Its Application to Geodesy and Astronomy, With Numerous Examples, Dublin: Hodges, Figgis, & Co., Grafton-St. London: Longmans, Green, & Co., (1889). $ \href{https://www.survivorlibrary.com/library/a_treatise_on_spherical_trigonometry_and_its_application_to_geodesy_and_astronomy_1889.pdf}{\mbox{[Web]}} $
4. [4] A.A. Ungar, Analytic Hyperbolic Geometry in N Dimensions, An Introduction, CRC Press, (2015). $ \href{https://doi.org/10.1201/b17858}{\mbox{[CrossRef]}} $
5. [5] K. Satˆo, Two centroids of spherical or hyperbolic triangle: the point trisecting the area and the concurrent point of arcs bisecting the area, Hiroshima Math. J., to appear.
6. [6] D.V. Alekseevskij, E.B. Vinberg and A.S. Solodovnikov, Geometry of Spaces of Constant Curvature, In: Vinberg, E.B. (eds) Geometry II. Encyclopaedia of Mathematical Sciences, Springer-Verlag, 29 (1993). $ \href{https://doi.org/10.1007/978-3-662-02901-5_1}{\mbox{[CrossRef]}} $
7. [7] K. Satˆo, Incenters, circumcenters, and orthocenters of simplices of Euclidean spaces, spheres, and hyperbolic spaces: The interiorness and their polarity, Proceedings of NACA-ICOTA2019 II (2019), 213-227. $\href{http://www.ybook.co.jp/book/NACA-ICOTA2019/naca-icota-II-p213/HTML5/}{\mbox{[Web]}} $