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 $\delta \mathrm{\ne }\frac{1}{2}$. 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 $P$.