The Reverse Kakeya Problem
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.
Kakeya problem
convex
isodynamic point
turning
6:1-6:13
Regular Paper
Sang Won
Bae
Sang Won Bae
Sergio
Cabello
Sergio Cabello
Otfried
Cheong
Otfried Cheong
Yoonsung
Choi
Yoonsung Choi
Fabian
Stehn
Fabian Stehn
Sang Duk
Yoon
Sang Duk Yoon
10.4230/LIPIcs.SoCG.2018.6
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.
http://dx.doi.org/10.1007/PL00009337
A. S. Besicovitch. Sur deux questions de l'intégrabilité. Journal de la Société des Math. et de Phys., II, 1920.
A. S. Besicovitch. On Kakeya’s problem and a similar one. Math. Zeitschrift, 27:312-320, 1928.
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.
A. DePano, Yan Ke, and J. O’Rourke. Finding largest inscribed equilateral triangles and squares. In Proc. 25th Allerton Conf. Commun. Control Comput., 1987.
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.
I. Laba. From harmonic analysis to arithmetic combinatorics. Bulletin (New Series) of the AMS, 45:77-115, 2008.
G. Pál. Ein Minimumproblem für Ovale. Math. Ann., 83:311-319, 1921.
T. Tao. From rotating needles to stability of waves: Emerging connections between combinatorics, analysis and PDE. Notices of the AMS, 48:297-303, 2001.
T. Wolff. Recent work connected with the Kakeya problem. In H. Rossi, editor, Prospects in Mathematics. American Math. Society, 1999.
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode