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Minimum Convex Partitions and Maximum Empty Polytopes

  • Adrian Dumitrescu
  • Sariel Har-Peled
  • Csaba D. Tóth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)

Abstract

Let S be a set of n points in d-space. A convex Steiner partition is a tiling of conv(S) with empty convex bodies. For every integer d, we show that S admits a convex Steiner partition with at most ⌈(n − 1)/d⌉ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension d ≥ 3. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position.

Establishing a tight lower bound for the maximum volume of a tile in a Steiner partition of any n points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is ω(1/n) in any dimension d ≥ 2. Here we give a (1 − ε)-approximation algorithm for computing the maximum volume of an empty convex body amidst n given points in the d-dimensional unit box [0,1] d .

Keywords

Convex Hull Convex Body General Position Unit Cube Convex Polygon 
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 2012

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Sariel Har-Peled
    • 2
  • Csaba D. Tóth
    • 3
    • 4
  1. 1.Computer ScienceUniversity of Wisconsin–MilwaukeeUSA
  2. 2.Computer ScienceUniversity of Illinois at Urbana–ChampaignUSA
  3. 3.Mathematics and StatisticsUniv. of CalgaryCanada
  4. 4.Comp. Sci.Tufts UniversityCanada

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