Article

Israel Journal of Mathematics

, Volume 188, Issue 1, pp 1-23

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 Technion-Israel Institute of Technology
  • , Simeon ReichAffiliated withDepartment of Mathematics, The Technion-Israel Institute of Technology

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Abstract

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 Curtis-Schori 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.