Uniqueness of Flotation and Buoyancy Surfaces for Convex Polytopes

TL;DR

This paper proves that convex polytopes are uniquely determined by their flotation and buoyancy surfaces for densities other than 1/2, establishing a unique geometric characterization.

Contribution

It establishes the uniqueness of convex polytopes based on their flotation and buoyancy surfaces for all densities except 1/2.

Findings

Convex polytopes are uniquely determined by their flotation surface for densities not equal to 1/2.

Convex polytopes are uniquely determined by their buoyancy surface for all densities in (0,1).

The results extend the understanding of geometric characterization of convex bodies.

Abstract

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 $P$.