Abstract: Croft, Falconer and Guy posed a series of questions generalizing Ulam's floating body problem, as follows.
Given a convex body K in R^3, we consider its plane sections with certain given properties,
(V): Plane sections which cut off a given constant volume
- Plane sections which have a given constant area
(I) Plane sections which have equal constant principal moments of inertia
Ulam's floating body problem is equivalent to problem (V,I): If all plane sections of the body K which cut off equal volumes have equal constant moments of inertial, must K be an Euclidean ball?
We give a negative answer to problem (V,A) following Ryabogin's counterexample to Ulam's floating body problem. We also give a positive answer to problem (A,I) in the class of bodies of revolution.
In addition, we prove several local fixed point results for the centroid body (the surface of buoyancy associated to Ulam's floating body problem when the density of K is 1/2).
This is joint work with Gulnar Aghabalayeva and Chase Reuter.