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Title of Journal: Math Ann

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Abbravation: Mathematische Annalen

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Springer-Verlag

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DOI

10.1016/0148-9062(77)90926-3

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1432-1807

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Zone diagrams in Euclidean spaces and in other nor

Authors: Akitoshi Kawamura Jiří Matoušek Takeshi Tokuyama
Publish Date: 2011/11/17
Volume: 354, Issue: 4, Pages: 1201-1221
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Abstract

Zone diagrams are a variation on the classical concept of Voronoi diagrams Given n sites in a metric space that compete for territory the zone diagram is an equilibrium state in the competition Formally it is defined as a fixed point of a certain “dominance” map Asano Matoušek and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in the Euclidean plane and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary finite dimension and more generally in a finitedimensional normed space with a smooth and rotund norm The proof is considerably simpler than that of Asano et al We also provide an example of nonuniqueness for a norm that is rotund but not smooth Finally we prove existence and uniqueness for two point sites in the plane with a smooth but not necessarily rotund norm

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Other Papers In This Journal:
  1. Estimates of the Green function for X-elliptic operators
  2. Fibred multilinks and singularities $${f \overline g}$$
  3. The NSLUC property and Klee envelope
  4. Hölder estimates for parabolic p ( x , t )-Laplacian systems
  5. Ricci flow on surfaces with cusps
  6. A non-laminar dynamical Green current
  7. Artin characters, Hurwitz trees and the lifting problem
  8. Generic non-selfadjoint Zakharov–Shabat operators
  9. Strichartz estimates via the Schrödinger maximal operator
  10. Examples of relative deformation spaces that are not locally connected
  11. Polynomial approximation of Berkovich spaces and definable types
  12. Prographes sylvestres et groupes profinis presque libres
  13. A critical radius for unit Hopf vector fields on spheres
  14. Many triangulated 3-spheres
  15. New examples of c 0 -saturated Banach spaces
  16. Rational curves, Dynkin diagrams and Fano manifolds with nef tangent bundle
  17. Relatively weakly open sets in closed balls of Banach spaces, and real JB * -triples of finite rank
  18. The Bergman metric on complete Kähler manifolds
  19. Conformal dimension via subcomplexes for small cancellation and random groups
  20. Analytic torsion of Hirzebruch surfaces
  21. Déformations des cônes de vecteurs primitifs
  22. Radial entire solutions for supercritical biharmonic equations
  23. Weighted Calderón–Zygmund and Rellich inequalities in $$L^p$$
  24. On entropy, regularity and rigidity for convex representations of hyperbolic manifolds
  25. Rigidity of proper holomorphic maps between bounded symmetric domains
  26. On CR Paneitz operators and CR pluriharmonic functions
  27. Absence of wandering domains for some real entire functions with bounded singular sets
  28. Isomonodromic deformations of logarithmic connections and stability
  29. Pointwise equidistribution with an error rate and with respect to unbounded functions
  30. On the Frobenius stable part of Witt vector cohomology
  31. Quantitative mixing results and inner functions
  32. Configuration spaces, transfer, and 2-nodal solutions of a semiclassical nonlinear Schrödinger equation
  33. Spin curves and Scorza quartics
  34. Points rationnels des variétés de Shimura: un principe du “tout ou rien”
  35. On bounded and unbounded global solutions of a supercritical semilinear heat equation
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  37. On Oliver’s p -group conjecture: II
  38. Scattering in the energy space for the NLS with variable coefficients
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