However, there are an infinite number of 2-dimensional objects which have a constant distance between two parallel surfaces. Drawings and a video. The Reuleaux triangle is the simplest of these.
The Wankel engine is based on shape similar to the Reuleaux triangle .. but with the edges slightly flattened.
It is also possible to produce non-spherical 3-dimensional objects that will maintain a constant distance between two parallel surfaces. Solids of constant width provides photos and a video (or two).
The cross section of the Reuleaux tetrahedra (the volume created by the intersection of 4 spheres) does not have a constant thickness because the face to edge distance is slightly larger than the point to face difference. (About 2% larger.) However, if some of the edges are rounded a bit, the Reuleaux tetrahedra can be converted into a non-spherical solid of constant thickness.