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Farthest point map on a centrally symmetric convex polyhedron

  • Zili WangEmail author
Original Paper
  • 21 Downloads

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.

Keywords

Convex polyhedron Centrally symmetric Farthest point Geodesic Limit set 

Mathematics Subject Classification (2010)

52A15 37E99 

Notes

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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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Brown UniversityProvidenceUSA

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