Sang Won Bae, Sergio Cabello, Otfried Cheong, Yoonsung Choi, Fabian Stehn, Sang Duk Yoon
January 22, 2021
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 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.