Zone diagrams in Euclidean spaces and in other normed spaces

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F12%3A10125729" target="_blank" >RIV/00216208:11320/12:10125729 - isvavai.cz</a>

Result on the web

<a href="http://dx.doi.org/10.1007/s00208-011-0761-1" target="_blank" >http://dx.doi.org/10.1007/s00208-011-0761-1</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00208-011-0761-1" target="_blank" >10.1007/s00208-011-0761-1</a>

Result language

angličtina

Original language name

Zone diagrams in Euclidean spaces and in other normed spaces

Original language description

Zone diagrams are a variation on the classical concept of Voronoi diagrams. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain "dominance" map. Asano, Matouek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in the Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.

Czech name

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Czech description

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Type

J_x - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)

CEP classification

BA - General mathematics

OECD FORD branch

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Project

<a href="/en/project/1M0545" target="_blank" >1M0545: Institute for Theoretical Computer Science</a><br>

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2012

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Mathematische Annalen

ISSN

0025-5831

e-ISSN

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Volume of the periodical

354

Issue of the periodical within the volume

4

Country of publishing house

DE - GERMANY

Number of pages

21

Pages from-to

1201-1221

UT code for WoS article

000310830600001

EID of the result in the Scopus database

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