Farthest Point Map on a Centrally Symmetric Convex Polyhedron

TL;DR

This paper investigates the properties of the composition of the farthest point map with the antipodal map on centrally symmetric convex polyhedra, revealing convergence and geometric structure of the orbits.

Contribution

It introduces a detailed analysis of the composed map, showing it has no generalized periodic points and that all orbits converge to a set contained in finitely many hyperbolas.

Findings

The map has no generalized periodic points.

All orbits of the map converge.

The limit set is contained in a finite union of hyperbolas.

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 polyhedron, so that we can compose the farthest point map with the antipodal map. The purpose of this work is to study the properties of this composition. We show that: 1. the map has no generalized periodic points; 2. its limit point 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 in the article.