Algorithmica

, Volume 70, Issue 2, pp 152–170 | Cite as

A Generalization of the Convex Kakeya Problem

  • Hee-Kap Ahn
  • Sang Won Bae
  • Otfried Cheong
  • Joachim Gudmundsson
  • Takeshi Tokuyama
  • Antoine Vigneron
Article
  • 319 Downloads

Abstract

Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya’s problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal Θ(nlogn)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. 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 every rotated copy of G.

Keywords

Computational geometry Discrete geometry Algorithms Kakeya 

Notes

Acknowledgements

We thank Helmut Alt, Tetsuo Asano, Jinhee Chun, Dong Hyun Kim, Mira Lee, Yoshio Okamoto, János Pach, Günter Rote, and Micha Sharir for helpful discussions.

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Copyright information

© Springer Science+Business Media New York 2013

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.POSTECHPohangSouth Korea
  2. 2.Kyonggi UniversitySuwonSouth Korea
  3. 3.KAISTDaejeonSouth Korea
  4. 4.University of Sydney and NICTASydneyAustralia
  5. 5.Tohoku UniversitySendaiJapan
  6. 6.Geometric Modeling and Scientific Visualization CenterKing Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia

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