The Ulam floating body problem asks whether a homogeneous convex body in three-dimensional Euclidean space
that floats in equilibrium in every orientation must be a closed
ball. It is Problem 19 of the Scottish Book,
proposed by Ulam, and also appears in Ulam (1960, p. 38). The density of the
surrounding fluid is normalized to 1, and the density 
of the body satisfies 
.
For each orientation, a plane cuts off the submerged fraction 
of the body's volume. Equilibrium requires the center
of mass of the body and the geometric centroid
of the submerged portion to lie on a line perpendicular
to this plane (Ryabogin 2022).
Ryabogin (2022), whose result was first announced in 2021, gave a negative answer at 
. For every dimension 
, he constructed a strictly convex
body of revolution in 
that is not a closed ball
but floats in equilibrium in every orientation with half its volume
submerged. In three dimensions, this is a solid
of revolution. The counterexamples are small smooth perturbations of a closed
ball and are not centrally symmetric.
The symmetry qualification is essential. At density 
, a centrally symmetric convex body floating in every orientation must be a closed ball (Schneider 1970, Ryabogin 2022).
The convex floating body construction instead takes an intersection of half-spaces obtained by cutting caps of prescribed volume from a given convex body. It does not impose the equilibrium condition in Ulam's problem.
