In geometry, a surface of constant width is a convex form whose width, measured by the distance between two opposite parallel planes touching its boundary, is the same regardless of the direction of those two parallel planes.
More generally, any compact convex body D has one pair of parallel supporting planes in a given direction.
Contrary to common belief the Reuleaux tetrahedron is not a surface of constant width.
However, there are two different ways of smoothing subsets of the edges of the Reuleaux tetrahedron to form Meissner tetrahedra, surfaces of constant width.
Among all surfaces of revolution with the same constant width, the one with minimum volume is the shape swept out by a Reuleaux triangle rotating about one of its axes of symmetry,[1] while the one with maximum volume is the sphere.