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
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