Zone and double zone diagrams in abstract spaces

• Autorzy: Daniel Reem , Simeon Reich
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 06B23: Complete lattices, completions
• 54E35: Metric spaces, metrizability
• 47H10: Fixed-point theorems
• 51K99: None of the above, but in this section

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2009
• Tom: 115
• Numer: 1
• Strony: 129-145

Daty

wydano

2009

Twórcy

autor

Daniel Reem

• Department of Mathematics, The Technion - Israel Institute of Technology, 32000 Haifa, Israel

autor

Simeon Reich

• Department of Mathematics, The Technion - Israel Institute of Technology, 32000 Haifa, Israel

Identyfikatory

DOI

10.4064/cm115-1-11