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)

Abstract

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.

References

  1. 1.
    Ahn, H.-K., Cheong, O.: Aligning two convex figures to minimize area or perimeter. Algorithmica, http://dx.doi.org/10.1007/s00453-010-9466-1
  2. 2.
    Besicovitch, A.S.: Sur deux questions de l’intégrabilité. Journal de la Société des Math. et de Phys. II (1920)Google Scholar
  3. 3.
    Besicovitch, A.S.: On Kakeya’s problem and a similar one. Math. Zeitschrift 27, 312–320 (1928)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Besicovitch, A.S.: The Kakeya problem. American Math. Monthly 70, 697–706 (1963)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Besicovitch, A.S.: On fundamental geometric properties of plane line sets. Journal of the London Math. Society 39, 441–448 (1964)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bezdek, K., Connelly, R.: Covering curves by translates of a convex set. American Math. Monthly 96, 789–806 (1989)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bezdek, K., Connelly, R.: The minimum mean width translation cover for sets of diameter one. Beiträge zur Algebra und Geometrie 39, 473–479 (1998)MathSciNetMATHGoogle Scholar
  8. 8.
    Bourgain, J.: Harmonic analysis and combinatorics: How much they contribute to each other? In: Arnold, V.I., Atiyah, M., Lax, P., Mazur, B. (eds.) Mathematics: Frontiers and Perspectives, pp. 13–32. American Math. Sociaty (2000)Google Scholar
  9. 9.
    Chakerian, G.D.: Sets of constant width. Pacific J. of Math. 19, 13–21 (1966)MathSciNetMATHGoogle Scholar
  10. 10.
    Fisher, B.: On a problem of Besicovitch. American Math. Monthly 80(7), 785–787 (1973)MATHCrossRefGoogle Scholar
  11. 11.
    Kakeya, S.: Some problems on maximum and minimum regarding ovals. Tohoku Science Reports 6, 71–88 (1917)MATHGoogle Scholar
  12. 12.
    Konagurthu, A., Whisstock, J., Stuckey, P., Lesk, A.: MUSTANG: a multiple structural alignment algorithm. Proteins 64, 559–574 (2006)CrossRefGoogle Scholar
  13. 13.
    Laba, I.: From harmonic analysis to arithmetic combinatorics. Bulletin (New Series) of the AMS 45(1), 77–115 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Ohmann, D.: Extremalprobleme für konvexe Bereiche der euklidischen Ebene. Math. Zeitschrift 55, 346–352 (1952)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Pál, G.: Ein Minimumproblem für Ovale. Math. Ann. 83, 311–319 (1921)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Perron, O.: Über einen Satz von Besicovitch. Math. Zeitschrift 28, 383–386 (1928)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Rademacher, H.A.: On a theorem from Besicovitch. In: Szego, G. (ed.) Studies in Mathematical Analysis and Related Topics: Essays in Honor of George Pòlya, pp. 294–296. Stanford University Press (1962)Google Scholar
  18. 18.
    Schoenberg, I.J.: On certain minima related to the Besicovitch-Kakeya problem. Mathematika 4(27), 145–148 (1962)MathSciNetGoogle Scholar
  19. 19.
    Schoenberg, I.J.: On the Besicovitch–Perron solution of the Kakeya problem. In: Szego, G. (ed.) Studies in Mathematical Analysis and Related Topics: Essays in Honor of George Pòlya, pp. 359–363. Stanford University Press (1962)Google Scholar
  20. 20.
    Tao, T.: From rotating needles to stability of waves: Emerging connections between combinatorics, analysis and PDE. Notices of the AMS 48(3), 297–303 (2001)Google Scholar
  21. 21.
    Fejes Tóth, L.: On the densest packing of convex disks. Mathematika 30, 1–3 (1983)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Toussaint, G.: Solving geometric problems with the rotating calipers. In: Proceedings of IEEE MELECON, pp. 1–4 (1983)Google Scholar
  23. 23.
    Vigneron, A.: Geometric optimization and sums of algebraic functions. In: Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms, pp. 906–917 (2010)Google Scholar
  24. 24.
    Wolff, T.: Recent work connected with the Kakeya problem. In: Rossi, H. (ed.) Prospects in Mathematics. American Math. Sociaty (1999)Google Scholar

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

Personalised recommendations