Ulam's Problem 19 from the Scottish Book asks:
*is a solid of uniform density which will float in water in every
position a sphere?*

Assuming that the density of water is $1$, one can 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 will discuss this and related problems suggested by Croft, Falconer and Guy.

Meeting id: 975 0965 2005