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

TL;DR

This paper investigates smooth surfaces of constant width in three-dimensional space, establishing integral inequalities, characterizations, and methods for constructing explicit examples with specific symmetries.

Contribution

It introduces new integral inequalities, characterizations, and construction techniques for C^2-smooth constant width surfaces in Euclidean 3-space.

Findings

Proved an integral inequality for constant width surfaces.

Showed the volume-to-width ratio decreases when shrinking along normals.

Constructed explicit smooth constant width surfaces with tetrahedral symmetry.

Abstract

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.