Uniqueness of Flotation and Buoyancy Surfaces for Convex Polytopes

We prove that a convex polytope $P \subset \mathbb{R}^d$, $d \ge 2$, of uniform density $\delta \in (0,1)$ floating in a liquid of density $1$, is uniquely determined by its surface of flotation $P_{[\delta]}$ whenever $\delta \neq \tfrac{1}{2}$. Analogously, we show that the buoyancy surface $\mathcal{C}_\delta P$ of a convex polytope $P$ with prescribed density $\delta \in (0,1)$ uniquely determines $P$.