Voronoi diagrams appear in many areas in science and technology and have numerous applications. They have been the subject of extensive investigation during the last decades. Roughly speaking, they are a certain decomposition of a given space into cells, induced by a distance function and by a tuple of subsets called the generators or the sites. Consider the following question: does a small change of the sites, e.g., of their position or shape, yield a small change in the corresponding Voronoi cells? This question is by all means natural and fundamental, since in practice one approximates the sites either because of inexact information about them, because of inevitable numerical errors in their representation, for simplification purposes and so on, and it is important to know whether the resulting Voronoi cells approximate the real ones well. The traditional approach to Voronoi diagrams, and, in particular, to (variants of) this question, is combinatorial. However, it seems that there has been a very limited discussion in the geometric sense (the shape of the cells), mainly an intuitive one, without proofs, in Euclidean spaces. We formalize this question precisely, and then show that the answer is positive in the case of Rd, or, more generally, in (possibly infinite dimensional) uniformly convex normed spaces, assuming there is a common positive lower bound on the distance between the sites. Explicit bounds are given, and we allow infinitely many sites of a general form. The relevance of this result is illustrated using several pictures and many real-world and theoretical examples and counterexamples.

The bisector of two nonempty sets P and Q in a metric space is the set of all points with equal distance to P and to Q. A distance k-sector of P and Q, where k ≥ 2 is an integer, is a (k-1)-tuple (C1, C2, ..., Ck-1) such that Ci is the bisector of Ci-1 and Ci+1 for every i= 1, 2, ..., k-1, where C0 = P and Ck = Q. This notion, for the case where P and Q are points in Euclidean plane, was introduced by Asano, Matousek, and Tokuyama, motivated by a question of Murata in VLSI design. They established the existence and uniqueness of the distance trisector in this special case. We prove the existence of a distance k-sector for all k and for every two disjoint, nonempty, closed sets P and Q in Euclidean spaces of any (finite) dimension, or more generally, in proper geodesic spaces (uniqueness remains open). The core of the proof is a new notion of k-gradation for P and Q, whose existence (even in an arbitrary metric space) is proved using the Knaster-Tarski fixed point theorem, by a method introduced by Reem and Reich for a slightly different purpose.