## Tbilisi Mathematical Journal

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

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

Brendan Guilfoyle and Wilhelm Klingenberg

#### 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**

Received: 19 February 2008

Revised: 11 February 2009

Accepted: 17 March 2009

First available in Project Euclid: 12 June 2018

**Permanent link to this document**

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

Secondary: 52A38: Length, area, volume [See also 26B15, 28A75, 49Q20]

**Keywords**

Convex geometry constant width line congruence

#### Citation

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