It is obvious that spheres have a constant height, independent of the orientation.
More surprising are so called Meissner bodies, which have the same properties. A special case of a Meissner body is a rotated Reuleaux triangle, which has the smallest volume of all Meissner bodies (in contrast to a sphere with the biggest volume).
This model is hollow, with additional inner structure for higher stability. The height is 4 cm. A glass plate can "roll" on top of 3 rotated Reuleaux triangles without moving up and down. Ideal toy for mathematics and physics demonstrations.