On C$^2$-smooth Surfaces of Constant Width

Details On C$^2$-smooth Surfaces of Constant Width On C$^2$-smooth Surfaces of Constant Width

2007-04-24

arxiv.org/abs/0704.3248v1

Mathematics

Differential Geometry

14 pages AMS-LATEX, 5 figures

Scientific paper

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.

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