In this paper, we obtain a number of results for C²-smooth surfaces of constant width in Euclidean 3-space E³-. In particular, we establish an integral inequality for constant width surfaces. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shrinking it along its normal lines. We also give a characterization of surfaces of constant width that have rational support function.
Our techniques, which are complex differential geometric in nature, allow us to construct explicit smooth surfaces of constant width in E³, and their focal sets. They also allow for easy construction of tetrahedrally symmetric surfaces of constant width.

Tbilisi Mathematical Journal, Vol. 2 (2009), pp. 1-17