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Piercing Translates and Homothets of a Convex Body

  • Adrian Dumitrescu
  • Minghui Jiang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5757)

Abstract

According to a classical result of Grünbaum, the transversal number \(\tau({\mathcal F})\) of any family \({\mathcal F}\) of pairwise-intersecting translates or homothets of a convex body C in ℝ d is bounded by a function of d. Denote by α(C) (resp. β(C)) the supremum of the ratio of the transversal number \(\tau({\mathcal F})\) to the packing number \(\nu({\mathcal F})\) over all families \({\mathcal F}\) of translates (resp. homothets) of a convex body C in ℝ d . Kim et al. recently showed that α(C) is bounded by a function of d for any convex body C in ℝ d , and gave the first bounds on α(C) for convex bodies C in ℝ d and on β(C) for convex bodies C in the plane. In this paper, we show that β(C) is also bounded by a function of d for any convex body C in ℝ d , and present new or improved bounds on both α(C) and β(C) for various convex bodies C in ℝ d for all dimensions d. Our techniques explore interesting inequalities linking the covering and packing densities of a convex body. Our methods for obtaining upper bounds are constructive and lead to efficient constant-factor approximation algorithms for finding a minimum-cardinality point set that pierces a set of translates or homothets of a convex body.

Keywords

Lattice Point Convex Body Lattice Packing Symmetric Convex Symmetric Convex Body 
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 2009

Authors and Affiliations

  • Adrian Dumitrescu
    • 1
  • Minghui Jiang
    • 2
  1. 1.Department of Computer ScienceUniversity of Wisconsin-MilwaukeeUSA
  2. 2.Department of Computer ScienceUtah State UniversityLoganUSA

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