A negative answer to Ulam's Problem 19 from the Scottish Book

TL;DR

The paper constructs a counterexample in three and higher dimensions of a convex body of uniform density that floats in equilibrium in all orientations, disproving Ulam's Problem 19 about solids of uniform density being spheres.

Contribution

It provides the first known counterexamples of convex bodies of uniform density that are not spheres yet float in equilibrium in all orientations, in dimensions three and higher.

Findings

Existence of non-spherical convex bodies of uniform density that float in all orientations.

Construction of such bodies as bodies of revolution with density 1/2.

Extension of the result to all dimensions d ≥ 3.

Abstract

We give a negative answer to Ulam's Problem 19 from the Scottish Book asking {\it is a solid of uniform density which will float in water in every position a sphere?} Assuming that the density of water is $1$, we show that there exists a strictly convex body of revolution $K\subset {\mathbb R^3}$ of uniform density $\frac{1}{2}$, which is not a Euclidean ball, yet floats in equilibrium in every orientation. We prove an analogous result in all dimensions $d\ge 3$.