We prove a generalization of Pál's conjecture from 1921: 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 Ω(m n2) 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.
Funding statement: A preliminary version of this work appeared in Proc. 34th International Symposium on Computational Geometry (SoCG 2018). SB is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07042755). SC is supported by the Slovenian Research Agency, program P1-0297. OC and YC are supported by the ICT R&D program of MSIT/IITP [IITP-2015-0-00199].
This research was started during the Korean Workshop on Computational Geometry 2017, organized by Tohoku University in Zao Onsen. We thank all workshop participants for the helpful discussions. We also thank Xavier Goaoc and Helmut Alt for helpful discussions.
Communicated by: M. Joswig
 P. K. Agarwal, N. Amenta, M. Sharir, Largest placement of one convex polygon inside another. Discrete Comput. Geom. 19 (1998), 95–104. MR1486639 Zbl 0892.68101Search in Google Scholar
 A. S. Besicovitch, Sur deux questions de l'intégrabilité des functions. J. Soc. Phys.–Math. (Perm) 2 (1919), 105–123.Search in Google Scholar
 A. S. Besicovitch, On Kakeya's problem and a similar one. Math. Z. 27 (1928), 312–320. MR1544912 JFM 53.0713.01Search in Google Scholar
 J. Bourgain, Harmonic analysis and combinatorics: how much may they contribute to each other? In: Mathematics: frontiers and perspectives 13–32, Amer. Math. Soc. 2000. MR1754764 Zbl 0963.42001Search in Google Scholar
 N. A. De Pano, Y. Ke, J. O'Rourke, Finding largest inscribed equilateral triangles and squares. In: Proc. 25th Allerton Conf. Commun. Control Comput. Univ. Illinois, 869–878 (1987).Search in Google Scholar
 S. Kakeya, Some problems on maxima and minima regarding ovals. Science Reports, Tôohoku Imperial Univ. 6 (1917), 71–88. JFM 46.1119.02Search in Google Scholar
 I. aba, From harmonic analysis to arithmetic combinatorics. Bull. Amer. Math. Soc. (N.S.) 45 (2008), 77–115. MR2358378 Zbl 1160.11004Search in Google Scholar
 J. Pál, Ein Minimumproblem für Ovale. Math. Ann. 83 (1921), 311–319. MR1512015 JFM 48.0838.02Search in Google Scholar
 T. Tao, From rotating needles to stability of waves: emerging connections between combinatorics, analysis, and PDE. Notices Amer. Math. Soc. 48 (2001), 294–303. MR1820041 Zbl 0992.42002Search in Google Scholar
 T. Wolff, Recent work connected with the Kakeya problem. In: Prospects in mathematics (Princeton, NJ, 1996) 129–162, Amer. Math. Soc. 1999. MR1660476 Zbl 0934.42014Search in Google Scholar
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