Zone and double zone diagrams in abstract spaces

Daniel Reem, and Simeon Reich. "Zone and double zone diagrams in abstract spaces." Colloquium Mathematicae 115.1 (2009): 129-145. <http://eudml.org/doc/283908>.

@article{DanielReem2009,
abstract = {A zone diagram of order n is a relatively new concept which was first defined and studied by T. Asano, J. Matoušek and T. Tokuyama. It can be interpreted as a state of equilibrium between n mutually hostile kingdoms. Formally, it is a fixed point of a certain mapping. These authors considered the Euclidean plane with finitely many singleton-sites and proved the existence and uniqueness of zone diagrams there. In the present paper we generalize this concept in various ways. We consider general sites in m-spaces (a simple generalization of metric spaces) and prove several existence and (non)uniqueness results in this setting. In contrast with previous works, our (rather simple) proofs are based on purely order-theoretic arguments. Many explicit examples are given, and some of them illustrate new phenomena which occur in the general case. We also re-interpret zone diagrams as a stable configuration in a certain combinatorial game, and provide an algorithm for finding this configuration in a particular case.},
author = {Daniel Reem, Simeon Reich},
journal = {Colloquium Mathematicae},
keywords = {dominance region; fixed point; -space; metric space; trisector; zone diagram},
language = {eng},
number = {1},
pages = {129-145},
title = {Zone and double zone diagrams in abstract spaces},
url = {http://eudml.org/doc/283908},
volume = {115},
year = {2009},
}

TY - JOUR
AU - Daniel Reem
AU - Simeon Reich
TI - Zone and double zone diagrams in abstract spaces
JO - Colloquium Mathematicae
PY - 2009
VL - 115
IS - 1
SP - 129
EP - 145
AB - A zone diagram of order n is a relatively new concept which was first defined and studied by T. Asano, J. Matoušek and T. Tokuyama. It can be interpreted as a state of equilibrium between n mutually hostile kingdoms. Formally, it is a fixed point of a certain mapping. These authors considered the Euclidean plane with finitely many singleton-sites and proved the existence and uniqueness of zone diagrams there. In the present paper we generalize this concept in various ways. We consider general sites in m-spaces (a simple generalization of metric spaces) and prove several existence and (non)uniqueness results in this setting. In contrast with previous works, our (rather simple) proofs are based on purely order-theoretic arguments. Many explicit examples are given, and some of them illustrate new phenomena which occur in the general case. We also re-interpret zone diagrams as a stable configuration in a certain combinatorial game, and provide an algorithm for finding this configuration in a particular case.
LA - eng
KW - dominance region; fixed point; -space; metric space; trisector; zone diagram
UR - http://eudml.org/doc/283908
ER -