Homogeneous spheres float stably in all possible orientations (Ulam 1960), but it has never been proved that no other homogeneous body shares this property (Gilbert 1991).
When the two above shapes have uniform density 0.5 they, like the uniform sphere of density 1, will float in a liquid in any orientation without tending to rotate (Mauldin 1982, Wells 1991, Gilbert 1991).
References
Auerbach, H. "Sur un problème de M. Ulam concernant l'équilibre des corps folttants." Studia Math. 7, 121-122, 1938.
Bassanini, P. and Bulgarelli, U. "An A-Priori Bound Concerning the Nonlinear Motion of Floating Bodies." Bollettino U. M. I. 8A, 141-149, 1994.
Croft, H. T.; Falconer, K. J.; and Guy, R. K. "The Floating Body Problem." §A6 in Unsolved Problems in Geometry. New York: Springer-Verlag, pp. 19-20, 1991.
Falconer, K. J. "Applications of a Result on Spherical Integration to the Theory of Convex Sets." Amer. Math. Monthly 90, 690-693, 1983.
Gilbert, E. N. "How Things Float." Amer. Math. Monthly 98, 201-216, 1991.
John, F. "On the Motion of Floating Bodies. I." Comm. Pure Appl. Math. 2, 13-57, 1949.
John, F. "On the Motion of Floating Bodies. II: Simple Harmonic Motions." Comm. Pure Appl. Math. 2, 45-101, 1950.
Mauldin, R. D. (Ed.). Problem 19 in The Scottish Book: Math at the Scottish Cafe. Boston: Birkhäuser, 1982.
Montejano, L. "On a Problem of Ulam Concerning Characterization of the Sphere." Studies Appl. Math. 53, 243-248, 1974.
Schneider, R. "Functional Equations Connected with Rotations and Their Geometric Applications." L'Enseign. Math. 16, 297-305, 1970.
Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 153-154, 1999.
Tietze, H. "Über die Anzahl der stabilen Ruhelagen eines Würfels." Elem. Math. 7, 97-100, 1948.
Ulam, S. M. A Collection of Mathematical Problems. New York: Interscience, p. 38, 1960.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 81, 1991.
