On \(C^{2}\)-smooth surfaces of constant width (Q991570)

The Blaschke Lebesgue theorem states that of all plane sets of given constant width the Reuleaux triangle has least area. The present paper is devoted to investigations in the case \(n=3\). Comparing the volume of a surface with constant width with the volume of the round sphere of the same width gives a functional which is proved to increase, when going to a parallel surface along the outer normals. So, obviously, one has to shrink the surface along the normals to get a better result. To keep convexity, the shrinking process has to be stopped when the surface touches the set of focal points. In case of a rational support function explicit examples with constant width are constructed.