  A number of results for C$^2$-smooth surfaces of constant width in Euclidean
3-space ${\mathbb{E}}^3$ are obtained. In particular, an integral inequality
for constant width surfaces is established. 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 ${\mathbb{E}}^3$,
and their focal sets. They also allow for easy construction of tetrahedrally
symmetric surfaces of constant width.
