Abstract
The farthest point map sends a point in a compact metric space to the set of points farthest from it. We focus on the case when this metric space is a convex centrally symmetric polyhedral surface, so that we can compose the farthest point map with the antipodal map. The purpose of this work is to study the properties of their composition. We show that: 1. the map has no generalized periodic points; 2. its limit set coincides with its generalized fixed point set; 3. each of its orbit converges; 4. its limit set is contained in a finite union of hyperbolas. We will define some of these terminologies later.
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Alexandrov, A. D.: Convex polyhedra, Translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze and A. B. Sossinsky. With comments and bibliography by V. A. Zalgaller and appendices by L. A. Shor and Yu. A. Volkov, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2005. xii+539 pp. ISBN: 3-540-23158-7
Aronov, B., O’Rourke, J.: Nonoverlap of the star unfolding, ACM symposium on computational geometry (North Conway, NH, 1991). Discret. Comput. Geom. 8(3), 219–250 (1992)
Rouyer, J.: On antipodes on a convex polyhedron. Adv. Geom. 5(4), 497–507 (2005)
Rouyer, J.: On antipodes on a convex polyhedron II. Adv. Geom. 10(3), 403–417 (2010)
Thurston, W.P.: The Geometry and Topology of Three-Manifolds. Princeton Mathematics Department, Princeton (1979)
Vîlcu, C.: On two conjectures of Steinhaus. Geom. Dedicata 79(3), 267–275 (2000)
Vîlcu, C.: Properties of the farthest point mapping on convex surfaces. Rev. Roum. Math. Pures Appl. 51(1), 125–134 (2006)
Vîlcu, C., Zamfirescu, T.: Multiple farthest points on Alexandrov surfaces. Adv. Geom. 7(1), 83–100 (2007)
Zamfirescu, T.: Extreme points of the distance function on a convex surface. Trans. Am. Math. Soc. 350(4), 1395–1406 (1998)
Acknowledgements
I would like to thank my adviser Richard Schwartz for many helpful discussions on the problem and feedbacks on the key ideas of proof. The Java program used in this paper is modified from his original program to compute the farthest point map on a regular octahedron. I am especially grateful to Joël Rouyer, who gives extensive feedbacks on this article, including (and not limited to) comments on the overall structure of this article and suggestions of some better proofs. I would also like to thank him and Costin Vîlcu for providing many of the references.
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Wang, Z. Farthest point map on a centrally symmetric convex polyhedron. Geom Dedicata 204, 73–96 (2020). https://doi.org/10.1007/s10711-019-00445-1
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DOI: https://doi.org/10.1007/s10711-019-00445-1