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Covering and Packing with Spheres by Diagonal Distortion in ℝn

  • Herbert Edelsbrunner
  • Michael Kerber
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6570)

Abstract

We address the problem of covering ℝ n with congruent balls, while minimizing the number of balls that contain an average point. Considering the 1-parameter family of lattices defined by stretching or compressing the integer grid in diagonal direction, we give a closed formula for the covering density that depends on the distortion parameter. We observe that our family contains the thinnest lattice coverings in dimensions 2 to 5. We also consider the problem of packing congruent balls in ℝ n , for which we give a closed formula for the packing density as well. Again we observe that our family contains optimal configurations, this time densest packings in dimensions 2 and 3.

Keywords

Packing covering spheres balls cubes lattices n-dimensional Euclidean space 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Herbert Edelsbrunner
    • 1
    • 2
    • 3
  • Michael Kerber
    • 1
  1. 1.IST Austria (Institute of Science and Technology Austria)KlosterneuburgAustria
  2. 2.Departments of Computer Science and of MathematicsDuke UniversityDurhamUSA
  3. 3.Geomagic, Research Triangle ParkUSA

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