  We prove that the largest convex shape that can be placed inside a given
convex shape $Q \subset \mathbb{R}^{d}$ in any desired orientation is the
largest inscribed ball of $Q$. The statement is true both when "largest" means
"largest volume" and when it means "largest surface area". The ball is the
unique solution, except when maximizing the perimeter in the two-dimensional
case.
