How to compute the rank of a Delaunay polytope

  • Authors:

  • Mathieu Dutour Sikirić;Viatcheslav Grishukhin

  • Affiliations:

  • Rudjer Boskovic Institute, Zagreb, Croatia;CEMI, Russian Academy of Sciences, Moscow, Russia

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2007

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Abstract

Roughly speaking, the rank of a Delaunay polytope is its number of degrees of freedom. In [M. Deza, M. Laurent, Geometry of Cuts and Metrics, Springer Verlag, Berlin, Heidelberg, 1997], a method for computing the rank of a Delaunay polytope P, using the hypermetrics related to P, is given. Here a simpler more efficient method, which uses affine dependencies instead of hypermetrics, is given. This method is applied to the classical Delaunay polytopes: cross-polytopes and half-cubes. Then, we give an example of a Delaunay polytope, which does not have any affine basis.