The Reverse Kakeya Problem

Authors Sang Won Bae, Sergio Cabello, Otfried Cheong, Yoonsung Choi, Fabian Stehn, Sang Duk Yoon



PDF
Thumbnail PDF

File

LIPIcs.SoCG.2018.6.pdf
  • Filesize: 0.58 MB
  • 13 pages

Document Identifiers

Author Details

Sang Won Bae
Sergio Cabello
Otfried Cheong
Yoonsung Choi
Fabian Stehn
Sang Duk Yoon

Cite AsGet BibTex

Sang Won Bae, Sergio Cabello, Otfried Cheong, Yoonsung Choi, Fabian Stehn, and Sang Duk Yoon. The Reverse Kakeya Problem. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 6:1-6:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SoCG.2018.6

Abstract

We prove a generalization of Pál's 1921 conjecture that if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously through 360° inside Q. We also prove a lower bound of Omega(m n^{2}) on the number of combinatorially distinct maximal placements of a convex m-gon P in a convex n-gon Q. This matches the upper bound proven by Agarwal et al.
Keywords
  • Kakeya problem
  • convex
  • isodynamic point
  • turning

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. P. K. Agarwal, N. Amenta, and M. Sharir. Largest placement of one convex polygon inside another. Discrete & Computational Geometry, 19:95-104, 1998. URL: http://dx.doi.org/10.1007/PL00009337.
  2. A. S. Besicovitch. Sur deux questions de l'intégrabilité. Journal de la Société des Math. et de Phys., II, 1920. Google Scholar
  3. A. S. Besicovitch. On Kakeya’s problem and a similar one. Math. Zeitschrift, 27:312-320, 1928. Google Scholar
  4. J. Bourgain. Harmonic analysis and combinatorics: How much may they contribute to each other? In V. I. Arnold, M. Atiyah, P. Lax, and B. Mazur, editors, Mathematics: Frontiers and Perspectives, pages 13-32. American Math. Society, 2000. Google Scholar
  5. A. DePano, Yan Ke, and J. O’Rourke. Finding largest inscribed equilateral triangles and squares. In Proc. 25th Allerton Conf. Commun. Control Comput., 1987. Google Scholar
  6. S. Kakeya. Some problems on maxima and minima regarding ovals. The Science Report of the Tohoku Imperial University, Series 1, Mathematics, Physics, Chemistry, 6:71-88, 1917. Google Scholar
  7. I. Laba. From harmonic analysis to arithmetic combinatorics. Bulletin (New Series) of the AMS, 45:77-115, 2008. Google Scholar
  8. G. Pál. Ein Minimumproblem für Ovale. Math. Ann., 83:311-319, 1921. Google Scholar
  9. T. Tao. From rotating needles to stability of waves: Emerging connections between combinatorics, analysis and PDE. Notices of the AMS, 48:297-303, 2001. Google Scholar
  10. T. Wolff. Recent work connected with the Kakeya problem. In H. Rossi, editor, Prospects in Mathematics. American Math. Society, 1999. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail