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| Presentation | 2010-01-25 11:30 Zone Diagrams in Euclidean Spaces and in Other Normed Spaces Akitoshi Kawamura (Univ. of Toronto.), Takeshi Tokuyama (Tohoku Univ.), Jiri Matousek (Charles Univ./ETH Zurich)  COMP2009-42 |
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| Abstract | (in Japanese) | (See Japanese page) |
| (in English) | Zone diagram is a variation on the classical concept of a Voronoi diagram. 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, Matousek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in 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. |
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| Keyword | (in Japanese) | (See Japanese page) |
| (in English) | Tarski fixed point theorem / zone diagrams / / / / / / | |
| Reference Info. | IEICE Tech. Rep., vol. 109, no. 391, COMP2009-42, pp. 23-28, Jan. 2010. | |
| Paper # | COMP2009-42 | |
| Date of Issue | 2010-01-18 (COMP) | |
| ISSN | Print edition: ISSN 0913-5685 Online edition: ISSN 2432-6380 | |
| Copyright and reproduction |
All rights are reserved and no part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Notwithstanding, instructors are permitted to photocopy isolated articles for noncommercial classroom use without fee. (License No.: 10GA0019/12GB0052/13GB0056/17GB0034/18GB0034) | |
| Download PDF | COMP2009-42 |
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| Conference Information | |
| Committee | COMP |
| Conference Date | 2010-01-25 - 2010-01-25 |
| Place (in Japanese) | (See Japanese page) |
| Place (in English) | Nishijin Plaza, Kyushu University |
| Topics (in Japanese) | (See Japanese page) |
| Topics (in English) | |
| Paper Information | |
| Registration To | COMP |
| Conference Code | 2010-01-COMP |
| Language | Japanese |
| Title (in Japanese) | (See Japanese page) |
| Sub Title (in Japanese) | (See Japanese page) |
| Title (in English) | Zone Diagrams in Euclidean Spaces and in Other Normed Spaces |
| Sub Title (in English) | |
| Keyword(1) | Tarski fixed point theorem |
| Keyword(2) | zone diagrams |
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| 1st Author's Name | Akitoshi Kawamura |
| 1st Author's Affiliation | University of Toronto (Univ. of Toronto.) |
| 2nd Author's Name | Takeshi Tokuyama |
| 2nd Author's Affiliation | Tohoku University (Tohoku Univ.) |
| 3rd Author's Name | Jiri Matousek |
| 3rd Author's Affiliation | Charles University/Swiss Federal Institute of Technology Zurich (Charles Univ./ETH Zurich) |
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| Speaker | Author-1 |
| Date Time | 2010-01-25 11:30:00 |
| Presentation Time | 35 minutes |
| Registration for | COMP |
| Paper # | COMP2009-42 |
| Volume (vol) | vol.109 |
| Number (no) | no.391 |
| Page | pp.23-28 |
| #Pages | 6 |
| Date of Issue | 2010-01-18 (COMP) |
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