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Distributions of Points and Large Convex Hulls of k Points

  • Hanno Lefmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4041)

Abstract

We consider a variant of Heilbronn’s triangle problem by asking for fixed integers d,k ≥2 and any integer nk for a distribution of n points in the d-dimensional unit cube [0,1] d such that the minimum volume of the convex hull of k points among these n points is as large as possible. We show that there exists a configuration of n points in [0,1] d , such that, simultaneously for j = 2, ..., k, the volume of the convex hull of any j points among these n points is Ω( 1/n (j − − 1)/(1 + |d − − j + 1|)). Moreover, for fixed kd+1 we provide a deterministic polynomial time algorithm, which finds for any integer nk a configuration of n points in [0,1] d , which achieves, simultaneously for j = d+1, ..., k, the lower bound Ω( 1/n (j − − 1)/(1 + |d − − j + 1|)) on the minimum volume of the convex hull of any j among the n points.

Keywords

Convex Hull Minimum Volume Unit Cube SIAM Journal London Mathematical Society 
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 2006

Authors and Affiliations

  • Hanno Lefmann
    • 1
  1. 1.Fakultät für InformatikTU ChemnitzChemnitzGermany

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