A Generalization of the Convex Kakeya Problem

  • Hee-Kap Ahn
  • Sang Won Bae
  • Otfried Cheong
  • Joachim Gudmundsson
  • Takeshi Tokuyama
  • Antoine Vigneron
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7256)


We consider the following geometric alignment problem: Given a set of line segments in the plane, find a convex region of smallest area that contains a translate of each input segment. This can be seen as a generalization of Kakeya’s problem of finding a convex region of smallest area such that a needle can be turned through 360 degrees within this region. Our main result is an optimal Θ(n logn)-time algorithm for our geometric alignment problem, when the input is a set of n line segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then the optimum placement is when the midpoints of the segments coincide. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of any rotated copy of G.


Line Segment Convex Hull Convex Region Regular Hexagon Symmetric Convex 
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

  • Hee-Kap Ahn
    • 1
  • Sang Won Bae
    • 2
  • Otfried Cheong
    • 3
  • Joachim Gudmundsson
    • 4
  • Takeshi Tokuyama
    • 5
  • Antoine Vigneron
    • 6
  1. 1.POSTECHSouth Korea
  2. 2.Kyonggi UniversitySouth Korea
  3. 3.KAISTSouth Korea
  4. 4.University of SydneyAustralia
  5. 5.Tohoku UniversityJapan
  6. 6.KAUSTSaudi Arabia

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