Floating Body Problems in Two Dimensions

摘要

Stanislav Ulam asked if the sphere is the only object floating in neutral equilibrium in every orientation and a negative answer was provided recently. Here, several related problems are discussed. The same question is asked for two-dimensional objects whose centroid is pinned and it is demonstrated that the answer is similar to the case of freely floating bodies. We also discuss the minimal number of equilibria of homogenous planar floating objects (either freely or with pinned centroid) representing duals of Ulam's Floating Body Problem. The nonexistence of shapes with less than four equilibria is proven in special cases including infinitesimal perturbations of a circle, however the general question remains open. The paper is complemented with remarks on analogous problems in three dimensions; connections to the family of Four-Vertex theorems are also pointed out.