, Volume 188, Issue 1, pp 123
Zone diagrams in compact subsets of uniformly convex normed spaces
 Eva KopeckáAffiliated withInstitute of Mathematics, Czech Academy of SciencesInstitut für Analysis, Johannes Kepler Universität Email author
 , Daniel ReemAffiliated withDepartment of Mathematics, The TechnionIsrael Institute of Technology
 , Simeon ReichAffiliated withDepartment of Mathematics, The TechnionIsrael Institute of Technology
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A zone diagram is a relatively new concept which has emerged in computational geometry and is related to Voronoi diagrams. Formally, it is a fixed point of a certain mapping, and neither its uniqueness nor its existence are obvious in advance. It has been studied by several authors, starting with T. Asano, J. Matoušek and T. Tokuyama, who considered the Euclidean plane with singleton sites, and proved the existence and uniqueness of zone diagrams there. In the present paper we prove the existence of zone diagrams with respect to finitely many pairwise disjoint compact sites contained in a compact and convex subset of a uniformly convex normed space, provided that either the sites or the convex subset satisfy a certain mild condition. The proof is based on the Schauder fixed point theorem, the CurtisSchori theorem regarding the Hilbert cube, and on recent results concerning the characterization of Voronoi cells as a collection of line segments and their geometric stability with respect to small changes of the corresponding sites. Along the way we obtain the continuity of the Dom mapping as well as interesting and apparently new properties of Voronoi cells.
 Title
 Zone diagrams in compact subsets of uniformly convex normed spaces
 Journal

Israel Journal of Mathematics
Volume 188, Issue 1 , pp 123
 Cover Date
 201203
 DOI
 10.1007/s1185601100945
 Print ISSN
 00212172
 Online ISSN
 15658511
 Publisher
 The Hebrew University Magnes Press
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 Authors

 Eva Kopecká ^{(1)} ^{(2)}
 Daniel Reem ^{(3)}
 Simeon Reich ^{(3)}
 Author Affiliations

 1. Institute of Mathematics, Czech Academy of Sciences, Žitná 25, CZ11567, Prague, Czech Republic
 2. Institut für Analysis, Johannes Kepler Universität, Altenbergerstrasse 69, A4040, Linz, Austria
 3. Department of Mathematics, The TechnionIsrael Institute of Technology, 32000, Haifa, Israel