Year: 2009
Type: article
Abstract: In this paper, we obtain a number of results for $\mathrm{C}^2$-smooth surfaces of constant width in Euclidean 3-space ${\mathbb{E}}^3$. 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 ${\mathbb{E}}^3$, and their focal sets. They also allow for easy construction of tetrahedrally symmetric surfaces of constant width.
Authors: Brendan Guilfoyle, Wilhelm Klingenberg
Institution: Durham University
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2899947528