On C^{2}-smooth Surfaces of Constant WidthBrendan Guilfoyle, Wilhelm Klingenberg
In this paper, we obtain a number of results for C^{2}-smooth surfaces of constant width in Euclidean
3-space E^{3}-.
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 E^{3}, 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 |