Zone diagrams in compact subsets of uniformly convex normed spaces

Details Zone diagrams in compact subsets of uniformly convex normed spaces Zone diagrams in compact subsets of uniformly convex normed spaces

2010-02-18

arxiv.org/abs/1002.3583v2

Mathematics

Functional Analysis

18 pages, 6 figures; Israel Journal of Mathematics, to appear; slight weakening of the main theorem by adding the emanation pr

Scientific paper

10.1007/s11856-011-0094-5

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. Matousek 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.

Affiliated with

Also associated with

No associations

Rating

Zone diagrams in compact subsets of uniformly convex normed spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Zone diagrams in compact subsets of uniformly convex normed spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Zone diagrams in compact subsets of uniformly convex normed spaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-406605