(The) reverse Kakeya problem

Abstract

We prove a generalization of $P\acute{a}l's$ 1921 conjecture that if a convex P can be placed in any orientation inside a convex Q in the plane, then P can also be turned continuously through $360^{\circ}$ 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.