Arithmetic Discrete Hyperspheres and Separatingness

  • Christophe Fiorio
  • Jean-Luc Toutant
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4245)


In the framework of the arithmetic discrete geometry, a discrete object is provided with its own analytical definition corresponding to a discretization scheme. It can thus be considered as the equivalent, in a discrete space, of a Euclidean object. Linear objects, namely lines and hyperplanes, have been widely studied under this assumption and are now deeply understood. This is not the case for discrete circles and hyperspheres for which no satisfactory definition exists. In the present paper, we try to fill this gap. Our main results are a general definition of discrete hyperspheres and the characterization of the k-minimal ones thanks to an arithmetic definition based on a non-constant thickness function. To reach such topological properties, we link adjacency and separatingness with norms.


Topological Property Discrete Space Linear Object Thickness Function Discrete Object 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Christophe Fiorio
    • 1
  • Jean-Luc Toutant
    • 1
  1. 1.LIRMM – CNRS UMR 5506 – Université de Montpellier IIMontpellierFrance

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