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Delaunay Polytopes: Rank and Hypermetric Faces

  • Michel Marie Deza
  • Monique Laurent
Part of the Algorithms and Combinatorics book series (AC, volume 15)

Abstract

There is a natural notion of rank for hypermetric spaces. Namely, if (X, d) is a hypermetric space, then its rank rk(X,d) is defined as the dimension of the smallest face of the cone HYP(X) that contains d. The extremal cases when the rank of (X, d) or its corank is equal to 1 correspond, respectively, to the cases when d lies on an extreme ray or on a facet of HYP(X). Correspondingly, the rank rk(P) of a Delaunay polytope P is defined as the rank of its Delaunay polytope space (V(P),d (2)). Delaunay polytopes of rank 1 are called extreme; they are associated to hypermetrics lying on an extreme ray of the hypermetric cone. This notion of rank for a Delaunay polytope P has the following geometric interpretation: It coincides with the number of degrees of freedom one has when deforming P in such a way that the deformed polytope remains a Delaunay polytope; a precise formulation can be found in Theorem 15.2.5.

Keywords

Relative Interior Antipodal Point Small Face Leech Lattice Triangle Facet 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Michel Marie Deza
    • 1
    • 2
  • Monique Laurent
    • 1
    • 3
  1. 1.Département de Mathématiques et d’InformatiqueLaboratoire d’Informatique de l’Ecole Normale SupérieureParis Cedex 05France
  2. 2.Department of MathematicsMoscow Pedagogical State UniversityMoscowRussia
  3. 3.CWIAmsterdamThe Netherlands

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