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)


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.


Lattice Point Convex Body Lattice Packing Symmetric Convex Symmetric Convex Body 
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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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