## Tbilisi Mathematical Journal

### On $\mathrm{C}^2$-smooth Surfaces of Constant Width

#### 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.

#### Note

The authors would like to thank Peter Giblin for bringing this topic to their attention and Rolf Schneider for helpful discussions. Part of this work was supported by the Research in Pairs Programme of the Mathematisches Forschungsinstitut Oberwolfach, Germany.

#### Article information

Source
Tbilisi Math. J., Volume 2 (2009), 1-17.

Dates
Revised: 11 February 2009
Accepted: 17 March 2009
First available in Project Euclid: 12 June 2018

https://projecteuclid.org/euclid.tbilisi/1528768838

Digital Object Identifier
doi:10.32513/tbilisi/1528768838

Mathematical Reviews number (MathSciNet)
MR2574869

Zentralblatt MATH identifier
1206.53008

Subjects
Primary: 53A05: Surfaces in Euclidean space
Guilfoyle, Brendan; Klingenberg, Wilhelm. On $\mathrm{C}^2$-smooth Surfaces of Constant Width. Tbilisi Math. J. 2 (2009), 1--17. doi:10.32513/tbilisi/1528768838. https://projecteuclid.org/euclid.tbilisi/1528768838