The reverse Kakeya problem

Abstract

We prove a generalization of Pal'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 degrees inside Q. We also prove a lower bound of Omega(mn(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.