Zone diagrams in compact subsets of uniformly convex normed spaces

TL;DR

This paper proves the existence of zone diagrams in compact, convex subsets of uniformly convex normed spaces, extending prior results beyond Euclidean settings using fixed point theorems and geometric stability analysis.

Contribution

It establishes the existence of zone diagrams for multiple disjoint compact sites in uniformly convex normed spaces under mild conditions, generalizing previous Euclidean results.

Findings

Existence of zone diagrams in uniformly convex normed spaces.

Continuity of the Dom mapping.

New properties of Voronoi cells and their stability.

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. 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.