Planning, geometry, and complexity of robot motion
Planning, geometry, and complexity of robot motion
Epsilon geometry: building robust algorithms from imprecise computations
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
Numerical stability of algorithms for 2D Delaunay triangulations
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
The stability of the Voronoi diagram
Computational Mathematics and Mathematical Physics
Space-efficient approximate Voronoi diagrams
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Linear onesided stability of MAT for weakly injective 3D domain
Proceedings of the seventh ACM symposium on Solid modeling and applications
A Replacement for Voronoi Diagrams of Near Linear Size
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Almost-Delaunay simplices: nearest neighbor relations for imprecise points
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Stability of persistence diagrams
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Simple Adaptive Mosaic Effects
SIBGRAPI '05 Proceedings of the XVIII Brazilian Symposium on Computer Graphics and Image Processing
A sampling theory for compact sets in Euclidean space
Proceedings of the twenty-second annual symposium on Computational geometry
Zone Diagrams: Existence, Uniqueness, and Algorithmic Challenge
SIAM Journal on Computing
On centroidal voronoi tessellation—energy smoothness and fast computation
ACM Transactions on Graphics (TOG)
Largest bounding box, smallest diameter, and related problems on imprecise points
Computational Geometry: Theory and Applications
Feature-Based Texture Synthesis and Editing Using Voronoi Diagrams
ISVD '09 Proceedings of the 2009 Sixth International Symposium on Voronoi Diagrams
Representing Dynamic Spatial Processes Using Voronoi Diagrams: Recent Developements
ISVD '09 Proceedings of the 2009 Sixth International Symposium on Voronoi Diagrams
An Algorithm for Computing Voronoi Diagrams of General Generators in General Normed Spaces
ISVD '09 Proceedings of the 2009 Sixth International Symposium on Voronoi Diagrams
Comparing Voronoi and Laguerre Tessellations in the Protein-Protein Docking Context
ISVD '09 Proceedings of the 2009 Sixth International Symposium on Voronoi Diagrams
Kinetic stable Delaunay graphs
Proceedings of the twenty-sixth annual symposium on Computational geometry
The Geometric Stability of Voronoi Diagrams with Respect to Small Changes of the Sites
Proceedings of the twenty-seventh annual symposium on Computational geometry
Visualization and analysis of protein structures using euclidean voronoi diagram of atoms
ICCSA'05 Proceedings of the 2005 international conference on Computational Science and Its Applications - Volume Part III
The Geometric Stability of Voronoi Diagrams with Respect to Small Changes of the Sites
Proceedings of the twenty-seventh annual symposium on Computational geometry
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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.